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FUNDAMENTAL   PRINCIPLES 

OF 

ELECTRIC  AND  MAGNETIC 
CIRCUITS 


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FUNDAMENTAL  PRINCIPLES 

OF 

ELECTRIC  AND  MAGNETIC 
CIKCUITS 


BY 

FRED  ALAN  FISH,  M.E.  IN  E.E. 

FROFESSOR-IN-CHARGE,  ELECTRICAL  ENGINEERING  DEPARTMENT, 
IOWA  STATE  COLLEGE;  FELLOW,  AMERICAN 

INSTITUTE  OF  ELECTRICAL  ENGINEERS 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,  INC. 

NEW  YORK:    239  WEST    39TH    STREET 

LONDON:  6  &  8  BOUVERIE  ST.,  E.  C.  4 

1920 


F  ' 

ginee 
Library 


COPYRIGHT,  1920,  BY  THE 
McGRAW-HiLL  BOOK  COMPANY,  INC. 


PREFACE 


THIS  book  has  been  written  as  an  introduction  to  the  study 
of  electric  power  machinery  and  transmission.  The  material 
contained  in  it  is  what  the  author  considers  to  be  the  vital  funda- 
mental principles.  It  is  intended  for  undergraduate  students  and 
therefore  does  not  go  as  deeply  into  the  physical  and  mathe- 
matical theory  of  electricity  and  magnetism  as  would  be  required 
for  graduate  study,  nor  does  it  include  all  the  possible  variations 
in  conditions  which  might  affect  the  application  of  the  principles 
as  laid  down.  These  may  be  brought  out  in  discussion  and  the 
student  taught  to  think  out  some  of  them  for  himself. 

The  author  desires  to  thank  Professors  F.  D.  Paine  and  F.  H. 
McClain  for  valuable  suggestions  in  the  preparation  of  the  book. 

F.  A.  FISH. 
AMES,  IOWA, 

June,  1920, 


CONTENTS 


PAGE 

Preface..  v 


CHAPTER  I 

FUNDAMENTALS 

1.  Introduction 1 

2.  Acceleration 1 

3.  Mass... 2 

4.  Force;  Weight;  Units  of  Force 3 

5.  Work;  the  Units,  Erg,  Joule  and  Foot-pound 5 

6.  Energy;  Difference  between  Work  and  Energy 5 

7.  Power;  Its  Relation  to  Work;  Units 6 


CHAPTER  II 

ELECTRICITY  AND  MAGNETISM 

8.  Electricity;  Potential;  Rate  of  Flow;  Current;  Ampere 8 

9.  Magnetism 10 

10.  Magnets;  Poles 10 

11.  Unit  Pole 11 

12.  Magnetic  Fields;  Lines  of  Force;  Unit  Strength  of  Field 12 

13.  Properties  of  Magnetic  Lines 13 

14.  Modern  Theory 14 

15.  Further  Laws  Concerning  Magnetic  Fields 15 

16.  Magnetic  Field  Around  a  Wire;  Positive  Direction 17 

17.  The  Solenoid;  Properties  of;  Direction  of  Flux  in 19 

18.  Action  of  Magnetic  Field  on  a  Wire  Carrying  Current;   Left-hand 

Rule 20 

19.  Unit  Current;  the  Abampere;  the  Ampere 21 

20.  International  Unit  of  Current 22 

21.  The  Coulomb;  the  Ampere-hour 23 

22.  Galvanometers 23 

vii 


viii  CONTENTS 

CHAPTER  III 

ELECTRIC  CIRCU.TS 

PAGE 

23.  Resistance;    Conductors;    Insulators;    Joule's  Law;    the  Unit  of 

Resistance;    the  Ohm;    Power  Consumed  in  Heating  a  Wire; 
International  Ohm 26 

24.  Ohm's  Law  and  Electromotive  Force;  Unit  of  Electromotive  Force; 

the  Volt;    Potential  Drop;    Rise  of  Potential;    Potential  Dis- 
tinguished from  Potential  Difference 27 

25.  Chemical  Sources  of  E.M.F.;  Voltaic  Cell;  Electrodes;  Action  in  a 

Cell  when  Current  Flows;   Reversibility  of  Cells;   Storage  Cells; 

the  Lead  Storage  Cell 29 

26.  Voltage  Relations  in  Battery  Circuits;   Internal  Resistance  Drop. .  31 

27.  Cells  in  Series 33 

28.  Cells  in  Parallel 34 

29.  Power  and  Energy  in  an  Electric  Circuit;    Definition  of  Electro- 

motive Force  as  Work  Done 36 

30.  The  Circular  Mil 39 

31.  Specific  Resistance;   Law  of  Resistance  as  Determined  by  Material 

and  Dimensions;    the  Mil-foot;    Resistivity  of  Copper;  Inter- 
national Standard 39 

32.  Effect  of  Temperature  on  Resistance;    Temperature  Coefficient; 

International  Standard  for  Copper;   Formulae  for  Resistance  and 

Temperature 40 

33.  Kirchhoff's  Laws 42 

34.  Resistances  in  Series 42 

35.  Resistances  in  Parallel;  Conductance 43 

36.  Series-Parallel  Circuits 44 

37.  Complex  Circuits;  Applications  of  Kirchhoff's  Laws 47 

38.  Wheatstone  Bridge;  Slide-wire  Bridge 51 

39.  Potentiometer 53 

40.  Ammeters  and  Voltmeters  for  Direct  Currents 55 


CHAPTER  IV 

ELECTROMAGNETISM 

41.  Flux-linkages  and  Electromotive  Force;    Definition  of  Linkage; 

Faraday's  Discovery;   Law  of  Induced  E.M.F.;   Reacting  Force 

in  a  Generator;  Reacting  Force  in  a  Motor;  Lenz'  Law 58 

42.  Relation  of  Induced  E.M.F.  to  Rate  of  Change  of  Linkages;  Funda- 

mental Equation  of  Induced  E.M.F.;   the  Abvolt;   Direction  of 

Induced  E.M.F.;  the  Right  hand  Rule 60 

43.  Work  Done  when  an  Electric  Wire  Cuts  a  Magnetic  Field 63 

44.  Number  of  Lines  of  Force  Issuing  from  a  Unit  Magnet  Pole 63 


CONTENTS  ix 

PAGE 

45.  Field  Intensity  around  a  Long  Straight  Wire 64 

46.  Force  Exerted  between  Two  Parallel  Wires 64 

47.  Field  Intensity  at  the  Center  of  a  Coil  of  Large  Radius 65 

48.  Magnetomotive  Force;   the  Measure  of  It;    the  Expression  for  It; 

the  Gilbert;    the  Magnitude  of  the  Gilbert;    the  Ampere-turn; 

its  Relation  to  the  Gilbert 66 

49.  Field  Intensity;  Magnetizing  Force 67 

50.  Flux  Density;  Permeability 68 

51.  Total  Flux  Produced  by  a  Coil 69 

52.  Reluctance;  Reluctivity 69 

53    Solution  of  Magnetic  Circuit  Problems;    Formula  for  Air  Gaps; 

Magnetization  Curves;    Determination  of  Ampere-turns  for  an 

Iron  Magnetic  Circuit 71 

54.  Series  Magnetic  Circuits;  Magnetic  Potential  Drop 73 

55.  Parallel  Magnetic  Circuits 75 

56.  Size  of  Wire  to  Produce  a  Given  M.M.F.  for  a  Given  Circuit 76 

57.  Field  Intensity  in  a  Solenoid;    Approximate  Formula 78 

58.  Magnetic  Leakage;   Leakage  Coefficient 78 

59.  Hysteresis;    Work  Done  in  Magnetizing  Iron;    Steinmetz'   Law; 

Hysteresis  Constants 78 

60.  Eddy  Currents  in  Iron 83 

61.  Pull  of  an  Electromagnet 85 

62.  Inductance;     Self-induction;     Unit    of    Inductance;     the    Henry; 

Millihenry;  Abhenry;  Brook's  Formula 86 

63.  Growth  of  Current  in  an  Inductive  Circuit 90 

64.  Decay  of  Current  in  an  Inductive  Circuit 91 

65.  Energy  of  a  Magnetic  Field 93 

66.  Inductance  of  Two  Long  Parallel  Wires 94 

67.  Skin  Effect. . 96 

68.  Mutual  Induction..  97 


CHAPTER  V 

ELECTROSTATIC  j 

69.  Electric  Charges 99 

70.  The  Electrostatic  Field;  Dielectrics 100 

71.  Electrostatic  Potential;    Unit  of  Electrostatic  Potential;    Electro- 

static Intensity 102 

72.  Capacity;  Dielectric  Constant;  Unit  of  Capacity;  the  Farad 103 

73.  Capacity  of  a  Parallel  Plate  Condenser 104 

74.  Condensers  in  Parallel  and  in  Series 105 

75.  Capacity  of  a  Transmission  Line 105 

76.  Charging  Current 107 

77.  Energy  of  a  Condenser 107 

78.  Distribution  of  Electrostatic  Intensity 108 


CONTENTS 


PAGE 

79.  Potential  Gradient 108 

80.  Losses  in  Dielectrics 110 

81.  Dielectric  Strength;  Corona 110 

82.  Charging  and  Discharging  a  Condenser  through  a  Resistance 112 

83.  Short-circuiting  Inductance  and  Capacity  in  Series 115 


CHAPTER  VI 
SINE  WAVE  ALTERNATING  CURRENTS 

84.  Definition  of  Alternating  Current 117 

85.  The  E.M.F.  and  Current  Equations;   Cycle;   Frequency;   Angular 

Velocity;    Electrical   Degrees;    Phase;    Positive  and   Negative 

Angles ,  117 

86.  Effective  and  Average  Values  of  Current  and  E.M.F 122 

87.  Current  and  E.M.F.  Waves  in  Resistance  Only 123 

88.  Current  and  E.M.F.  Waves  in  Indue  ance  Only 124 

89.  Current  and  E.M.F.  Waves  in  Capacity  Only 127 

90.  Vector  Representation  of  Alternating  Quantities 130 

91.  Current  and  E.M.F.  Relations  in  a  Circuit  Containing  Resistance, 

Inductance  and  Capacity;  Reactance;  Impedance 132 

92.  Effective  Resistance 134 

93.  Power  in  A.-C.  Circuits 135 

94.  Power  Factor;  Apparent  Power;  Reactive  Factor 138 

95.  Power  and  Reactive  Components  of  E.M.F 139 

96.  Power  and  Reactive  Components  of  Current;    Conductance;   Sus- 

ceptance;  Admittance 140 

97.  The  Symbolic  Method  of  Expressing  Vector  Quantities 141 

98.  Impedance  and  Admittance  as  Complex  Numbers 143 

99.  Impedances  in  Series 144 

100.  Electromotive  Forces  in  Series 145 

101.  Resonance  in  Series  Circuits 147 

102.  Impedances  in  Parallel 149 

103.  Currents  in  Parallel 150 

104.  Mixed  Circuits .  .  151 


CHAPTER  VII 
NON-HARMONIC  WAVES 

105.  Composition  of  Non-harmonic  Waves 154 

106.  The  Oscillograph 156 

107.  Analysis  of  a  Non-harmonic  Wave 157 

108.  Example ." 161 


CONTENTS  xi 

PAGE 

109.  Effective  Value  of  a  Non-harmonic  Wave 164 

110.  Peak  Factor 165 

111.  Average  Value  of  a  Non-harmonic  Wave. 165 

112.  Form  Factor 165 

113.  Power  in  Circuits  Carrying  Non-harmonic  Waves 165 

114.  Equivalent  Sine  Waves  and  Phase  Difference 167 

115.  Inductive  Reactance  with  Non-harmonic  Waves 167 

116.  Capacity  Reactance  with  Non-harmonic  Waves 169 


CHAPTER  VIII 
POLYPHASE  CURRENTS 

117.  Kirchhoff's  Laws  Applied  to  Alternating  Currents 171 

118.  Two-phase  Connections 171 

119.  Three-phase  Connections 176 

120.  Relation  of  Line  Voltages  to  Phase  Voltages  in  Three-phase  Delta- 

connected  Systems 177 

121.  Relation    of   Line    Currents    to    Phase    Currents   in    Three-phase 

Systems 178 

122.  Relation  of  Line  Voltage  to  Phase  Vol  ages  in  Three-phase  Y-con- 

nected  Systems 180 

123.  Power  in  Three-phase  Circuits 181 

124.  Power  Measurement  in  Three-phase  Circuits 182 

125.  Line  Drop  in  Three-phase  Circuits 189 


FUNDAMENTAL  PEINCIPLES 

OF 

ELECTRIC  AND  MAGNETIC  CIRCUITS 


CHAPTER   I 
FUNDAMENTALS 

1.  There  are  certain  fundamental  principles  and  ideas 
concerning  which  the  student  of  engineering  must  have  a 
very  clear  understanding  before  he  can  possibly  master  the 
more  complex  relations  and  processes  with  which  he  must 
deal  in  following  his  profession.     In  this  text,  it  is  assumed 
that  the  ideas  of  length,  time,  and  velocity  are  well  under- 
stood.    However,  on  account  of  their  great  importance, 
the  topics  of  acceleration,  mass,  force,  work,  energy  and 
power  will  be  discussed  in  review.     The  treatment  will  be 
brief  because  it  is  understood  that  these  subjects  have  been 
studied  before,  but  no  effort  should  be  spared  in  fixing  them 
firmly  in  mind. 

2.  Acceleration. — When  at   a  given   instant   a  body  is 
moving  at  such  a  rate  that  it  would  traverse  a  distance  of 
120  ft.  if  it  continued  to  move  at  the  same  rate  for  one 
second,  its  velocity  is  said  to  be  120  ft.  per  second.     Velocity 
expressed  in  this  way,  does  not  necessarily  mean  that  the 
body  will  travel  120  ft.  during  the  next  second,  but  that  it  is 
traveling  at  that  rate  at  the  given  instant.     The  rate  at 
which  the  velocity  of  a  body  changes  with  time  is  called  its 
acceleration.     If  at  a  given  instant  its  velocity  is  120  ft. 


2  ELECTRIC  AND  MAGNETIC  CIRCUITS 

per  second,  but  is  changing,  and  is  changing  at  such  a  rate 
that  at  the  end  of  one  second  its  velocity  would  be  130  ft. 
per  second,  its  acceleration  is  10  ft.  per  second  per  second. 
Again,  this  does  not  necessarily  mean  that  its  velocity  will 
be  10  ft.  per  second  greater  at  the  end  of  one  second,  but 
that  it  is  changing  at  that  rate  at  the  given  instant.  Accel- 
eration may  be  either  positive,  or  negative;  that  is,  the 
velocity  may  be  either  increasing  or  decreasing.  One 
of  the  most  common  examples  of  acceleration  is  that  due  to 
the  earth's  attraction,  or  the  acceleration  of  "  gravity." 
This  has  been  proven  by  experiment  to  be  a  constant  for 
all  kinds  of  bodies  for  any  given  place  on  the  earth's  surface, 
but  varies  slightly  with  latitude  and  altitude.  Its  value  is 
approximately  981  cm.  per  second  per  second  or  32.16  ft. 
per  second  per  second.  That  is,  the  velocity  of  a  falling 
body  increases  32.16  ft.  per  second  every  second  during  its 
fall. 

3.  Mass. — The  mass  of  a  body  is  defined  as  the  quantity 
of  matter  it  contains.  It  is  independent  of  volume,  shape  or 
chemical  composition.  It  is  also  entirely  independent  of  the 
force  of  gravity  or  weight,  although  the  earth's  attractive 
force  is  taken  advantage  of  in  comparing  masses.  If  two 
bodies  exactly  balance  each  other  when  suspended  one  from 
each  end  of  an  equal  arm  balance,  they  are  said  to  have 
equal  mass,  provided  no  forces  act  upon  them  besides  that 
of  gravity.  This  definition  is  entirely  arbitrary,  but  it  is 
found  that  mass  as  thus  defined  is  one  of  the  fundamental 
properties  of  matter.  This  method  of  comparing  masses 
is  based  on  the  law,  proven  true  only  by  experiment,  that 
the  earth's  attractive  force  at  a  given  point  always  pro- 
duces the  same  acceleration  on  any  body  regardless  of  the 
quantity  of  matter  it  contains.  There  is  no  other  reason 
for  believing  that  bodies  of  equal  weight  have  equal  mass. 
Any  arbitrary  portion  or  kind  of  matter  may  be  taken  as 
the  standard  of  mass,  and  in  fact,  an  arbitrary  piece  of  plat- 
inum-iridium  is  the  International  Standard.  It  is  known 
as  the  International  Kilogram.  A  more  common  standard 


FUNDAMENTALS  3 

or  unit,  is  the  gram,  which  is  the  1/1000  part  of  a  kilo- 
gram. In  England  and  in  the  United  States,  the  most 
common  standard  is  the  pound  and  is  represented  by  a 
piece  of  platinum  preserved  in  London. 

4.  Force. — Any  push,  pull,  pressure,  tension,  attrac- 
tion or  repulsion  which  changes  or  tends  to  change  the  state 
of  rest  or  motion  of  a  body  is  a  force.  Strictly  speaking, 
rest  is  a  state  of  motion;  however,  the  words,  "  rest  or 
motion "  are  used  here  to  avoid  misunderstanding.  A 
change  in  state  of  motion  includes  not  only  a  decrease  or  an 
increase  of  linear  velocity,  but  also  a  change  of  direction. 
We  have  learned,  originally  from  Newton,  that  so  long  as  a 
body  is  left  to  itself,  that  is,  not  acted  upon  by  any  outside 
influences,  it  will  continue  in  the  same  state  of  rest  or  motion. 
The  same  law  also  holds  when  outside  forces  act  upon  the 
body  provided  the  resultant  of  all  such  outside  forces  is  zero. 
Under  such  conditions  as  these,  a  body  is  said  to  be  in  a 
state  of  equilibrium.  The  principle  of  equilibrium  is  one 
which  should  be  thoroughly  mastered.  A  body  is  in  equi- 
librium when  it  is  at  rest,  or  when  it  is  moving  in  a  straight 
line  with  a  constant  velocity;  because  to  start  it  from  rest 
or  to  change  its  direction  or  velocity  requires  the  applica- 
tion of  a  force  which  is  not  balanced  by  a  force  in  the  oppo- 
site direction.  If  a  car  be  moving  along  a  straight  track 
at  uniform  speed,  it  is  in  equilibrium;  for  the  force  which 
drives  it  exactly  balances  the  forces  which  tend  to  stop  it; 
if  the  driving  force  were  to  exceed  the  restraining  forces  by 
ever  so  little,  the  car  would  be  accelerated;  if  the  driving 
forces  were  to  become  less  than  the  restraining  forces  by 
ever  so  little,"the  speed  of  the  car  would  decrease.  If  a 
man  pushes  against  a  body  at  rest,  but  is  unable  to  move  it, 
the  force  with  which  the  body  resists  is  equal  to  the  force 
with  which  the  man  pushes,  and  they  are  in  equilibrium; 
for  if  the  force  with  which  the  body  resisted  were  less  than 
that  with  which  the  man  pushed,  the  body  would  be  moved, 
that  is,  accelerated;  and  if  the  force  with  which  the  body 
resisted  were  greater  than  that  with  which  the  man  pushed, 


4  ELECTRIC  AND  MAGNETIC  CIRCUITS 

the  man  would  be  moved  backward,  that  is,  accelerated. 
We  have  learned  that  when  a  body  is  acted  upon  by  an  un- 
balanced outside  force,  it  will  be  accelerated  in  direct  pro- 
portion to  the  magnitude  of  the  force  so  acting.  Taking 
advantage  of  this  fact,  the  magnitude  of  a  force  is  measured 
fundamentally  by  the  acceleration  it  will  give  to  unit  mass. 
If  the  gram  is  taken  as  the  unit  of  mass  and  acceleration  is 
measured  in  centimeters  per  second  per  second,  then  the 
unit  of  force  is  called  the  dyne,  and  it  is  that  force  which 
will  give  to  a  mass  of  1  gm.  an  acceleration  of  1  cm.  per 
second  per  second.  This  unit  is  much  used  in  developing 
the  principles  of  electricity  and  magnetism.  If  the  pound 
is  taken  as  the  unit  of  mass,  and  acceleration  is  measured 
in  feet  per  second  per  second,  then  the  unit  of  force  is  called 
the  poundal.  It  is  the  force  which  will  give  to  a  mass  of 
1  Ib.  an  acceleration  of  1  ft.  per  second  per  second.  This 
unit,  however,  is  not  much  used.  From  the  definition  of 
force,  it  follows  that  the  force  required  to  give  a  body  any 
desired  acceleration  is  equal  to  the  product  of  its  mass 
and  the  acceleration. 

The  force  with  which  the  earth  attracts  a  mass  is 
called  its  weight;  and  since  the  acceleration  due  to  gravity 
is  a  constant  at  any  given  point,  it  follows  that  the  ratio  of 
weight  to  mass  is  a  constant  at  any  one  point.  That  is, 
the  weight  of  a  body  at  any  point  is  equal  to  its  mass  times 
the  value  of  the  acceleration  of  gravity  at  that  point.  For 
this  reason  the  force  equal  to  the  weight  of  unit  mass  is  very 
commonly  used  in  engineering  as  unit  force.  Since  the 
acceleration  due  to  gravity  varies  slightly  over  the  earth's 
surface  such  a  unit,  in  order  to  be  invariable,  must  be  based 
upon  some  standard  value  of  acceleration.  This  has  been 
agreed  upon  as  980.665  cms.  per  second  per  second,  or 
32.1739  ft.  per  second  per  second.  Unfortunately  the  same 
name  is  given  to  this  unit  of  force  as  to  the  unit  of  mass  and 
this  sometimes  leads  to  confusion.  For  example,  a  most 
common  unit  of  force  in  this  country  is  the  pound,  which 
is  also  the  English  unit  of  mass.  The  use  of  the  pound  as 


FUNDAMENTALS  5 

a  unit  of  force  also  leads,  in  Mechanics,  to  a  different  but 
unnamed  unit  of  mass.  The  acceleration  which  a  force  of 
1  Ib.  will  give  to  a  mass  of  1  Ib.  is  about  32.2  ft.  per  second 
per  second;  therefore  a  force  of  one  pound  would  give  an 
acceleration  of  one  foot  per  second  per  second  to  a  mass 
of  32.2  Ibs. ;  that  is,  using  the  pound  as  a  unit  of  force,  the 
mass  of  a  body  is  equal  to  its  weight  at  any  point  divided 
by  the  acceleration  of  gravity  at  that  point.  This,  of  course, 
is  a  constant,  as  it  should  be,  since  the  mass  of  a  given 
body  is  invariable.  The  pound  used  as  a  unit  of  force  is 
called  the  gravitational  unit.  There  is  nothing  inconsistent 
about  the  two  systems  of  units,  provided  -one  remembers 
that  the  force  of  gravity  (i.e.,  weight)  on  a  given  mass  varies 
slightly  from  place  to  place.  The  approximate  value  of  a 
force  of  1  Ib.  is  453.6x981  =445,000  dynes. 

5.  Work. — When  a  force  acting  upon  a  body  succeeds 
in  moving  it,  work  is  said  to  be  done  on  the  body.     The 
amount  of  work  done  upon  a  body  is  defined  as  the  product 
of  the  distance  moved  through   and  the  average  value  of 
the  force  which  causes  the  motion.     The  unit  of  work  is 
therefore  that  done  by  a  force  of  one  dyne  acting  through 
a  distance  of  1  cm.,  or  that  done  by  a  force  of  1  Ib.  acting 
through  a  distance  of  1  ft.,  depending  on  the  system  of 
units  employed.     The  first  unit  mentioned  is  sometimes 
called  the  centimeter-dyne,  but  is  more  commonly  called 
an  erg;  that  is,  of  course,  a  very  small  unit  and  a  more  com- 
mon one  is  the  joule ,  which  is  equal  to  10,000,000  ergs. 
The  second  unit  mentioned  is  known  as   the   foot-pound; 
no  single  word  has  been  coined  to  take  the  place  of  the 
compound  word.     Since  there  are  30.48  cm.  in  1  ft.  and 
445,000  dynes  in  1  Ib.,  there  are  30.48  X445,000  =  13,560,000 
ergs  in  1  ft.-lb.,  or  1.356  joules  in  1  ft.-lb. 

6.  Energy. — Nature  has  endowed  the  substances  of  the 
universe  with  certain  properties  by  which,  under  suitable 
conditions,  they  are  able  to  cause  motion  and  thus  to  do 
what  has  been  defined  above  as  work.     This  ability  to  do 
work  is  given  the  name  energy.     A  most  important  prin- 


6  ELECTRIC  AND  MAGNETIC  CIRCUITS 

ciple  which  the  engineer  must  never  forget  is  that  of  the 
Conservation  of  Energy.  This  principle  is  that  the  total 
amount  of  energy  in  the  universe  is  constant,  and  can 
neither  be  added  to  nor  subtracted  from.  This  law  is  not 
susceptible  of  mathematical  proof,  but  all  experience  leads 
to  the  conclusion  that  it  is  true,  and  it  is  to  be  accepted  as 
one  of  the  "  Articles  of  Faith,"  for  the  scientist  and  the 
engineer. 

However,  it  is  an  everyday  observation  that  energy 
can  be  and  repeatedly  is  transformed  from  one  of  several 
forms  to  others;  these  transformations  are  the  means  by 
which  all  processes  are  performed.  Many  of  these  trans- 
formations are  relatively  simple,  and  it  is  not  difficult  for  us 
to  form  a  mental  picture  of  the  processes  by  which  they  are 
accomplished;  others  are  more  obscure  and  we  are  obliged 
to  accept  the  manifestations  which  our  senses  perceive  and 
from  these  construct  a  more  or  less  fictitious  picture  of  the 
processes. 

The  definition  of  energy  as  the  ability  to  do  work  implies 
the  existence  of  force  within  the  substance  or  system  of 
substances  possessing  such  energy.  It  should  be  noted, 
however,  that  force  may  be  exerted  without  doing  work 
and  it  is  only  when  the  force  is  great  enough  to  overcome 
the  opposition  to  it  and  cause  motion  that  work  is  done. 
Work  cannot  be  done  without  the  transfer  or  transforma- 
tion of  energy,  and  the  amount  of  work  done  represents  a 
loss  of  energy  at  one  point,  or  of  some  kind,  and  the  gain 
of  an  equal  amount  at  other  points  or  of  some  other  kind. 
The  term  "  consumption  of  energy,"  therefore,  does  not 
mean  that  energy  is  destroyed,  but  that  it  is  only  changed 
to  some  other  place  or  kind  or  both.  It  follows  from  this 
that  energy  is  measured  in  the  same  units  as  work,  that  is, 
in  ergs,  joules,  or  foot-pounds;  but  distinction  between 
work  and  energy  must  be  carefully  kept  in  mind. 

7.  Power. — The  amount  of  work  done  by  a  force  in  over- 
coming resistance  through  a  given  distance  is  independent 
of  the  time  required  to  do  it.  A  force  of  100  Ibs.  may  move 


FUNDAMENTALS  7 

a  body  200  ft.  in  one  second  or  in  ten  seconds;  the  amount 
of  work  done  is  the  same  hi  both  cases.  But  the  rate  at 
which  work  is  done  is  generally  a  matter  of  great  importance, 
and  is  defined  as  power.  The  two  most  important  units  of 
power  are  the  watt  and  the  horse-power.  When  work  is 
done  at  the  rate  of  1  joule  (107  ergs)  per  second,  the  power  is 
1  watt ;  when  it  is  done  at  the  rate  of  550  ft.-lbs.  per  second 
or  33,000  ft.-lbs.  per  minute,  the  power  is  1  h.-p.  In  engi- 
neering it  is  always  much  more  convenient  to  measure  power 
than  to  measure  work.  Therefore,  when  the  work  done  or 
the  energy  "  consumed  "  in  a  given  time  is  required,  the 
power  is  measured,  the  time  recorded,  and  the  work  cal- 
culated as  the  product  of  power  and  time.  If  power  is 
measured  in  watts  or  horse-power,  and  time  in  seconds,  the 
work  will  be  expressed  in  joules  or  foot-pounds  respectively; 
but  since  the  second  is  so  small  a  unit,  it  is  common  to  use 
the  hour  as  a  unit  of  time  and  to  express  work  in  watt-hours 
or  horse-power-hours.  One  watt-hour  does  not  mean  that 
the  power  has  been  1  watt  and  the  time  one  hour,  but  that 
the  product  of  the  power  in  watts  and  the  time  in  hours  is 
1  watt-hour.  Thus,  100  watt-hours  may  be  used  in  0.5 
hour  at  a  rate  of  200  watts,  or  in  five  hours  at  a  rate  of 
20  watts.  Since  there  are  3600  seconds  in  one  hour  and 
1.356  joules  (or,  watt-seconds)  in  1  ft.-lb.,  there  will  be 
3600/1.3655=2655  ft.-lbs.  in  1  watt-hour. 


CHAPTER   II 
ELECTRICITY  AND   MAGNETISM 

8.  Electricity. — Let  a  strip  of  zinc  and  a  strip  of  copper 
be  placed  some  distance  apart  in  a  diluife  solution  of  sul- 
phuric acid.     It  will  be  found  that  there  exists  between  the 
two  strips  of  metal  a  kind  of  force  that  did  not  exist  before 
they  were  placed  in  the  solution.     For  instance,  if  they  are 
connected  together  outside  of  the  solution  by  a  piece  of 
wire,  the  temperature  of  the  wire  will  be  increased;    if 
the  wire  is  sufficiently  fine  it  will  become  so  hot  as  to  give 
off  light.     If  a  long  piece  of  wire  is  used  and  it  is  wound 
up  so  as  to  form  a  coil,  it  will  be  found  that  a  piece  of  iron 
will  be  attracted  toward  the  center  of  the  coil.     It  is  of 
course  impossible  for  us  to  see  the  mechanism  by  which 
these  acts  are  performed.    But  the  evidence  is  before  us 
that  out  of  the  chemical  energy  of  the  solution  a  different 
kind  of  energy  is  produced  which  is  transferred  by  some 
means  out  through  or  along  the  wire  and  some  of  it,  at  least, 
is  transformed  into  heat  energy.     The  energy  into  which 
the  chemical  energy  is  transformed  is  called  electrical  energy, 
the  something  by  means  of  which  this  energy  is  transferred 
out  through  the  wire  is  called  electricity,  and  an  electric 
current  is  said  to  flow  in  the  wire.     In  the  present  state 
of  our  knowledge  we  are  unable  to  define  electricity  any 
further  than  to  say  that  it  is  the  means  by  which  electrical 
energy  is  carried  from  one  place  to  another.     A  familiar 
example  of  this  kind  of  a  carrier  is  the  water,  which,  flowing 
in  pipes,  carries  with  it  the  energy  it  possesses  by  virtue  of 
having  been  put  under  a  pressure  higher  than  that  at  the 
point  toward  which  it  flows. 

Whatever  may  be  the  real  mechanism  within  the  cell 

8 


ELECTRICITY  AND  MAGNETISM  9 

described  above,  it  is  evident  that  one  of  the  terminals  is 
at  what  may  be  called  a  higher  electrical  pressure  than  the 
other.  In  electrical  terminology,  one  terminal  is  said  to 
have  a  higher  potential  than  the  other,  or  there  is  said  to  be  a 
potential  difference  between  them.  The  cause  of  this  poten- 
tial difference,  whatever  its  nature  may  be,  is  called  elec- 
tromotive force;  it  may  be  regarded  as  the  force  which 
causes  or  tends  to  cause  electricity  to  move  and  is  analogous 
to  the  difference  in  pressure  which  is  set  up  between  the 
intake  and  outlet  of  a  pump.  The  unit  of  electromotive 
force  is  called  a  volt;  the  magnitude  of  this  unit  will  be 
discussed  in  a  later  paragraph. 

We  have  spoken  of  electricity  as  a  carrier  of  electrical 
energy,  and  we  should,  therefore,  expect  to  have  to  deal 
with  it  quantitatively.  The  unit  quantity  of  electricity  is 
called  &  coulomb;  its  magnitude  will  be  taken  up  later.  A 
coulomb  of  electricity  may  be  thought  of  as  analogous  to  a 
cubic  foot  of  water.  In  hydraulics,  we  have  occasion  to 
deal  not  only  with  quantities  of  water,  but  also  with  the 
rate  of  flow  of  water,  as  for  instance,  in  pumping  machinery 
or  in  water-power  developments.  In  electrical  engineering 
we  have  much  less  occasion  for  dealing  with  the  quantity  of 
electricity  than  for  dealing  with  its  rate  of  flow,  since  elec- 
tricity is  used  only  for  transmitting  energy.  However,  the 
use  of  the  idea  of  quantity  is  necessary  for  the  development 
of  the  theory  of  certain  phenomena  and  apparatus. 

When  electrical  energy  is  being  transmitted  from  one 
place  to  another  by  means  of  wires,  electricity  is  said  to  be 
flowing  through  the  wires.  In  hydraulics,  the  rate  of  doing 
work  (that  is,  the  power)  is  equal  to  the  product  of  the  rate 
of  water  flow  and  the  difference  in  pressure  between  the  upper 
and  lower  levels;  similarly,  in  electrical  circuits  the  power 
developed  between  two  points  is  equal  to  the  product  of 
the  rate  at  which  electricity  is  flowing  and  the  difference  of 
potential  between  the  points.  In  hydraulics,  rate  of  water 
flow  is  expressed  in  cubic  feet  per  second.  In  electric 
circuits,  rate  of  flow  is  expressed  in  amperes;  one  ampere 


10  ELECTRIC  AND  MAGNETIC  CIRCUITS 

meaning  that  electricity  is  flowing  past  any  given  point  at 
the  rate  of  1  coulomb  per  second.  The  use  of  the  single 
word  to  express  the  idea  of  rate  should  be  carefully  noted; 
confusion  often  arises  from  thinking  of  the  ampere  as  a 
quantity  of  electricity.  We  frequently  use  the  word 
"  current  "  in  referring  to  a  flow  of  water,  and  say  that  the 
current  is  strong  or  the  current  is  weak  as  the  case  may  be, 
meaning  that  the  rate  of  flow  is  great  or  small;  in  like 
manner,  the  word  current  is  used  in  connection  with  flow  of 
electricity  and  means  its  rate  of  flow;  the  ampere  is  the  unit 
of  current,  and  not  the  unit  of  electricity. 

The  path  through  which  an  electric  current  .flows  is 
called  an  electric  circuit;  when  the  path  is  complete,  lead- 
ing out  from  the  source,  through  wires,  lamps,  motors,  and 
other  apparatus  and  back  to  and  through  the  source,  the 
circuit  is  said  to  be  closed.  A  steady  current  cannot  flow 
unless  a  closed  circuit  is  provided  for  it. 

9.  Magnetism. — Mention  has  been  made  of  the  effect 
on  a  piece  of  iron  when  it  is  brought  near  a  coil  -of  wire  carry- 
ing an  electric  current.    The  region  surrounding  a  wire 
which  is  carrying  an  electric  current  is  found  to  be  the  seat 
of  energy,  and  a  mechanical  force  is  found  to  be  exerted 
upon  certain  kinds  of  material,  particularly  iron  and  steel, 
if  they  are  brought  into  the  neighborhood  of  such  a  wire. 
This  kind  of  energy  is  called  magnetic  energy,  and  the  .force 
is  called  magnetic  force.     Magnetism  is  the  general  name 
given  to  the  condition  or  state  of  affairs  which  exists  in  such 
a  region,  and  the  region  itself  is  called  a  magnetic  field. 

10.  Magnets. — If  a  bar  of  steel  be  placed  inside  a  coil 
and  current  be  sent  through  the  coil,  it  will  be  found  on 
removing  the  bar  from  the  coil  that  the  bar  has  acquired 
and  retained  the  same  property  possessed  by  the  coil,  namely 
that  of  attracting  toward  it  pieces  of  iron  or  steel.     A  bar, 
rod  or  needle  of  steel  possessing  this  property  is  said  to  be 
magnetized  and  is  called  a  magnet.     The  magnetic  force 
exerted  by  a  magnet  is  found  to  be  by  far  the  greatest  at 
the  ends  of  the  axis  which  was  parallel  with  the  axis  of  the 


ELECTRICITY  AND  MAGNETISM  11 

coil  in  which  the  bar  or  rod  was  magnetized.  These  ends 
are  called  the  poles  of  the  magnet.  This  concentration  of 
the  magnetic  force  at  the  ends  of  a  magnet  is  much  more 
pronounced  in  long,  slim  magnets  than  in  short  thick 
ones. 

Either  end  of  a  bar  of  unmagnetized  iron  will  be  attracted 
to  either  pole  of  a  magnet;  but  if  the  bar  be  magnetized 
one  end  will  be  attracted  and  the  other  end  repelled.  The 
earth  is  found  to  be  in  a  permanently  magnetized  condition 
with  one  pole  located  near  the  geographic  north  pole  and 
the  other  near  the  geographic  south  pole.  If  a  magnetized 
bcir  or  needle  be  suspended  so  as  to  swing  freely  in  a  horizon- 
tal plane,  one  pole  will  point  north  and  one  south;  the  north- 
pointing  pole  of  a  magnet  is  called  its  north  pole  and  the 
south-pointing  pole  its  south  pole.  The  north  pole  of  a 
magnet  always  attracts  the  south  pole  of  another  magnet 
while  the  north  poles  of  any  two  magnets  or  their  south 
poles  always  repel  each  other;  that  is,  like  poles  repel  each 
other  and  unlike  poles  attract  each  other.  A  north  pole  is 
frequently  called  a  positive  pole  and  a  south  pole  a  negative 
pole.  The  force  exerted  between  the  two  poles  is  con- 
sidered to  be  positive  when  it  is  a  repulsive  force. 

11.  Unit  Pole. — It  is  found  to  be  impossible  to  produce 
a  north  pole  without  producing  a  south  pole  of  equal  strength. 
Nevertheless,  it  is  often  convenient  to  imagine  an  isolated 
magnet  pole,  the  other  pole  being  thought  of  as  so  far  away 
as  to  have  no  effect  at  the  point  under  consideration.  The 
strength  of  a  pole  is  measured  by  the  force  with  which  it 
attracts  or  repels  another  pole.  The  force  of  attraction 
or  repulsion  between  two  poles  has  been  experimentally 
proven  to  be  inversely  proportional  to  the  square  of  their 
distance  apart.  A  magnet  pole  is  said  to  have  unit  strength, 
or  is  said  to  be  a  unit  pole,  when  it  exerts  a  force  of  one 
dyne  on  another  pole  of  equal  strength  at  a  distance  of  I  cm. 
The  strength  of  any  given  pole  in  c.g.s.  units  is  therefore 
numerically  equal  to  the  force  in  dynes  which  it  exerts  on  a 
unit  pole  1  cm.  away.  In  general,  the  force  exerted  between 


12  ELECTRIC  AND  MAGNETIC  CIRCUITS 

any  two  poles  is  equal  to  the  product  of  their  strengths 
divided  by  the  square  of  their  distance  apart. 

In  building  up  our  system  of  electric  and  magnetic  units 
this  idea  of  a  unit  magnet  pole  was  used  as  the  starting  point. 
This  is  generally  regarded  as  having  been  unfortunate,  as 
it  involves  some  very  difficult  and  delicate  measurements, 
and  some  troublesome  constants.  However,  these  measure- 
ments have  been  made  and  legal  values  of  the  units  estab- 
lished so  that  reference  back  to  the  fundamental  measure- 
ments is  not  now  required. 

12.  Magnetic  Fields. — Any  region  in  which  a  magnetic 
force  would  be  exerted  on  a  magnet  pole  is  a  magnetic  field. 
The  strength  of  a  magnetic  field  is  measured  by  the  force 
exerted  upon  a  unit  magnet  pole.  A  magnetic  field  is  said 
to  have  unit  strength  at  a  given  point  when  it  exerts,  or 
would  exert,  a  force  of  1  dyne  on  a  unit  pole  placed  at  that 
point.  The  direction  of  a  magnetic  field  at  any  point  is 
the  direction  of  the  force  which  would  be  exerted  upon  an 
isolated  north  pole  placed  at  the  given  point.  A  magnetic 
field  is  conveniently  pictured  in  the  mind  by  imagining  lines 
drawn  through  the  field,  their  direction  at  every  point  being 
the  same  as  the  direction  of  the  force  at  that  point,  and 
their  density  at  any  point  representing  the  value  of  the 
force,  or  the  strength  of  the  field  at  that  point.  Such  lines 
are  called  lines  of  magnetic  force,  or  in  short,  lines  of  force. 
A  line  of  force  therefore  represents  the  path  which  would 
be  taken  by  a  magnet  pole  if  it  could  be  isolated  and  left 
free  to  move  in  a  magnetic  field.  A  field  of  unit  strength 
at  any  point  would  be  represented  by  one  line  of  force  per 
square  centimeter  of  cross-sectional  area  at  the  given  point, 
the  area  being  taken  in  a  plane  at  right  angles  to  the  direc- 
tion of  the  line  of  force.  Since  the  field  may  not  be  uniform, 
this  does  not  mean  that  each  square  centimeter  is  to  be 
thought  of  as  containing  one  line  of  force,  but  that  the  den- 
sity of  the  lines,  at  the  given  point,  is  equal  to  one  line  per 
square  centimeter.  The  strength  of  a  magnetic  field  is 
commonly  called  its  intensity,  and  unit  intensity  is  one  dyne 


ELECTRICITY  AND  MAGNETISM  13 

per  unit  pole.  The  product  of  the  intensity  and  any  given 
area,  at  right  angles  to  the  direction  of  the  field  and  in  which 
the  given  intensity  is  uniform,  is  called  the  magnetic  flux 
across  that  area.  Since  intensity  is  represented  by  the 
number  of  lines  of  force  per  unit  area,  the  flux  is  repre- 
sented by  the  total  number  of  lines  of  force  crossing  the 
given  area.  One  line  of  force  or  unit  magnetic  flux,  is 
called  a  maxwell.  The  path  along  which  magnetic  forces 
act  is  known  as  a  magnetic  circuit.  Experiment  has  proven 
that  every  magnetic  circuit  is  closed  upon  itself;  that  is,  if 
an  isolated  pole  be  imagined  to  start  moving  from  any  point 
in  a  magnetic  field  and  to  keep  moving  always  in  the  direc- 
tion of  the  force  action,  the  path  followed  will  lead  back  to 
the  starting  point,  coming  up  to  it  from  the  opposite  direc- 
tion from  which  the  pole  started,  and  passing,  on  the  way, 
through  the  source  of  the  magnetic  force.  Lines  of  force 
are  therefore  always  to  be  thought  of  as  closed  lines,  and  all 
the  lines  which  issue  from  one  pole  of  a  magnet  must  return 
to  the  other  pole  and  pass  through  the  magnet.  The  posi- 
tive direction  of  a  line  of  force  is  taken  to  be  that  in  which 
a  north  magnet  pole  would  be  urged  to  move;  that  is,  the 
lines  are  assumed  to  leave  a  magnet  by  its  north  pole  and 
enter  it  at  its  south  pole. 

13.  Properties  of  Magnetic  Lines. — While  it  is  probable 
that  the  magnetic  flux  has  no  motion  around  its  circuit, 
yet  it  is  very  convenient  to  consider  that  it  does  move.  In 
picturing  this,  a  line  of  force  is  to  be  thought  of  as  moving 
in  a  direction  parallel  with  itself  and  at  every  point  in  the 
direction  of  the  force  action  at  that  point.  It  is  also  neces- 
sary to  think  of  this  motion  as  without  friction,  since 
repeated  experiments  have  proven  that  no  energy  is  required 
to  maintain  a  magnetic  field.  Imagine  that  a  small  tube 
is  bent  around,  and  following  the  direction  of  the  force, 
is  closed  upon  itself,  and  encloses  within  it  a  continuous 
stretched  rubber  band;  suppose  that  by  some  means  this 
band  be  made  to  move  along  through  the  tube  without 
friction;  this  is  the  kind  of  motion  a  line  of  force  may  be 


14  ELECTRIC  AND  MAGNETIC  CIRCUITS 

pictured  as  having.  The  stretched  rubber  band  is  used  in 
this  illustration  to  typify  the  tension  and  the  tendency  to 
shorten  themselves,  which  lines  of  force  are  experimentally 
proven  to  possess.  A  further  important  property  which 
lines  of  force  are  found  to  possess  is  that  they  repel  each 
other  when  in  the  same  direction  and  attract  each  other  when 
in  opposite  directions.  This  property  is  sometimes  pictured 
by  imagining  that  the  lines  are  whirling  with  high  velocity 
about  their  longitudinal  axes,  and,  being  of  elastic  material, 
they  push  against  each  other  when  whirling  in  the  same 
direction  and  pull  together  when  whirling  in  opposite  direc- 
tions. It  is  also  to  be  noted  that  lines  of  force  cannot  cross 
each  other. 

14.  Modern  Theory. — The  modern  working  theory  of 
magnetism  is  that  the  molecules  of  all  magnetizable  sub- 
stances are  permanent  magnets  which  in  general  point  at 
random  in  all  directions.  When  a  number  of  these  molecular 
magnets  come  under  the  influence  of  a  magnetizing  force, 
such  as  that  produced  by  an  electric  current  or  by  a  magnet 
of  appreciable  size,  it  is  supposed  that  they  swing  around 
under  the  influence  of  this  force,  so  that  their  north  poles 
point  in  one  general  direction  and  their  south  poles  in  the 
opposite  direction.  Inasmuch  as  a  directed  magnetic  field 
can  be  produced  in  air  or  in  a  vacuum,  it  must  be  supposed 
that  the  molecules  of  the  ether  are  likewise  in  a  permanently 
magnetized  condition.  A  considerable  number  of  material 
substances  are  found  to  be  magnetizable,  but  among  them 
iron  stands  out  as  one  which  is  magnetizable  to  a  far  greater 
degree  than  all  others.  While  a  magnetic  field  produced  in 
air  immediately  disappears  when  the  directive  force  is 
removed,  it  is  found  that  under  certain  conditions  as  to  its 
composition  a  piece  of  iron  will  retain  magnetic  properties 
to  a  marked  degree  after  the  directive  force  is  removed. 
This  is  called  residual  magnetism  and  is  accounted  for  by 
assuming  a  kind  of  molecular  friction  to  exist  in  the  iron 
which  tends  to  prevent  the  molecules  from  returning  to 
their  random  positions.  In  annealed  iron  this  friction  is 


ELECTRICITY  AND  MAGNETISM  15 

very  small,  in  cast  iron  and  mild  steel  it  is  greater,  and  in 
hardened  steel  it  is  very  great.  Conversely,  it  requires  a 
much  greater  magnetizing  force  to  magnetize  a  piece  of 
hardened  steel  to  a  given  degree  than  to  magnetize  a  piece 
of  annealed  iron  to  the  same  degree.  When  a  large  perma- 
nent magnet  is  brought  near  a  small  one,  the  north  pole  of 
the  former  will  naturally  attract  the  south  pole  of  the  latter, 
but  if  the  north  pole  of  the  small  one  be  forcibly  held  to 
the  north  pole  of  the  large  one,  the  small  one  may  have  its 
polarity  entirely  reversed  by  the  stronger  directive  force 
of  the  large  one.  If  either  end  of  a  bar  of  soft  iron  be  placed 
near  the  north  pole  of  a  permanent  magnet,  all  the  molecular 
south  poles  of  the  soft  bar  will  be  attracted  toward  the  north 
pole  of  the  magnet,  the  ends  of  the  bar  as  a  whole  will  become 
polarized,  and  the  bar  is  said  to  be  magnetized  by  induction. 
15.  Further  Laws  Concerning  Magnetic  Fields. — Since 
lines  of  magnetic  force  are  closed  lines,  and  all  lines  must 
pass  through  the  source  of  the  magnetic  field,  it  fol- 
lows that  the  lines  which  pass  through  a  magnet  must 
return  to  it  through  the  air  outside.  The  cross-sectional 
area  of  the  return  path  is  infinite  as  compared  with  the 
cross-soctional  area  of  the  magnet,  and  as  soon  as  the 
lines  leave  the  north  pole  of  a  magnet,  they  spread  out 
due  to  the  repulsion  between  them  and  the  intensity  of  the 
field  becomes  weaker  and  weaker  as  the  distance  from  the 
pole  increases.  As  has  been  stated,  lines  of  force  possess 
both  longitudinal  tension  and  transverse  repulsion;  the 
tension  in  the  lines  tends  to  shorten  them  and  thus  to  bring 
them  closer  together,  while  the  repulsion  between  lines 
tends  to  put  them  farther  apart ;  therefore  the  configuration 
of  a  given  set  of  lines  of  force  will  be  determined  by  the  posi- 
tion of  equilibrium  between  these  two  forces.  These  ideas 
are  based  upon  the  very  important  principle  that  whenever 
two  or  more  magnetic  fields  are  brought  within  acting  dis- 
tance of  each  other,  they  will  tend  to  place  themselves  in 
such  a  position  that  their  paths  will  be  parallel  and  in  the 
same  direction,  and  as  short  as  possible.  This  law,  like  the 


16  ELECTRIC  AND  MAGNETIC  CIRCUITS 

law  of  gravity,  is  one  of  the  fundamental  principles  of  nature, 
and  it  is  the  principle  on  which  all  electric  motors  and  meters 
operate.  It  applies  whether  the  fields  are  produced  by 
magnets,  or  coils  carrying  electric  current,  or  a  combination 
of  the  two.  Its  application  to  the  simple  cases  of  attraction 
and  repulsion  may  be  illustrated  by  the  following  examples: 
In  Fig.  1,  let  A  be  a  permanent  magnet  and  B  be  either  a 


\ 


/ 


FIG.  1. 

permanent  magnet  or  a  piece  of  soft  iron;  in  the  latter  case 
the  iron  will  be  magnetized  by  induction  and  become  a  tem- 
porary magnet.  It  is  readily  seen  that  in  this  case  the 
longitudinal  tension  of  the  lines  which  are  common  to  the 
two  magnets  will  draw  the  magnets  toward  each  other  and 
also  the  direction  of  the  lines  not  common  to  the  two  mag- 
nets is  such  that  they  attract  each  other.  In  Fig.  2  where 


FIG.  2. 

the  N-pole  of  A  is  set  opposite  the  N-pole  of  B,  there  are  no 
lines  common  to  the  two  magnets  and  therefore  longitudinal 
tension  has  no  effect  in  this  case;  however,  the  two  sets  of 
lines  are  in  such  direction  with  reference  to  each  other  that 
the  resultant  force  is  that  of  repulsion.  In  Fig.  3,  where 


ELECTRICITY  AND  MAGNETISM 


17 


magnet  B  is  set  on  a  pivot  above  magnet  A}  the  repulsion 
between  the  lines  would  cause  B  to  swing  around  after 
which  longitudinal  tension  would  come  into  play  and  the 


\  ' 
\\  ! 

-  V    \    \ 


s 


FIG.  3. 


equilibrium  position  would  be  as  shown  in  Fig.  5.  Fig.  4 
shows  the  condition  (looking  down  on  the  magnets)  after  B 
has  swung  through  90°. 


FIG.  4. 


16.  The  Magnetic  Field  around  a  Wire. — When  a  wire  is 
carrying  a  current,  the  direction  of  the  magnetizing  force 
which  it  exerts  on  a  magnet  pole  depends  upon  the  direction 
in  which  the  current  is  flowing  through  the  wire.  Experi- 
ment has  proven  that  the  force  action  of  the  field  due  to  a 


18 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


long  straight  wire  is  at  right  angles  to  the  wire  and  at  every 
point  tangential  to  a  circle  drawn  through  the  point,  the 
center  of  the  circle  being  at  the  center  of  the  wire,  and  the 
plane  of  the  circle  being  at  right  angles  to  the  axis  of  the 


B 


t<H\     \ 
•^     \M./     7 


/  i 


FIG.  5. 

wire.  This  is  shown  in  Fig.  6,  where  the  wire  is  represented 
in  cross-section  at  the  center  of  the  figure,  the  axis  of  the 

wire  being  at   right   angles  to 

,,-'".'-     -~-"-I>x  the  plane  of  the  paper,  and  the 

dotted  circles  representing   the 
lines  of  force. 

To  illustrate  further,  suppose 
a  current  is  flowing  in  the  wire, 
WW,  Fig.  7a,  in  such  a  direction 
that  if  a  small  compass  needle 
\K^:;"::;'''V'V          be  placed  above  the  wire  as  at 

(/),  the  north  pole  will  be  de- 

FIG.  6.  fleeted  to  the  right ;   if  placed 

below  the  wire  as  at  (g)  it  will 

be  deflected  to  the  left;  that  is,  if  one  looks  along  the  wire 
in  the  direction  from  W  toward  W,  the  north  pole  of  the 
needle  will  always  point  in  a  clockwise  direction. 

It  is  desirable  to  agree  on  which  direction  shall  be  taken 
as  the  positive  direction  of  an  electric  current  in  a  wire. 
The  convention  which  has  been  universally  agreed  to  is  as 
follows :  If  when  looking  along  the  axis  of  a  wire  a  north,  or 
positive,  magnet  pole  tends  to  move  clockwise  around  the 


ELECTRICITY  AND  MAGNETISM 


19 


wire,  the  current  is  considered  as  flowing  away  from  the 
observer,  and  this  is  taken  as  the  positive  direction.  Con- 
versely, if  the  current  flows  away  from  an  observer  looking 
along  the  axis  of  the  wire  the  magnetic  field  is  considered  as 
directed  clockwise  around  the  wire.  When  looking  at  the 
cross-section  of  a  wire  as  in  Fig.  76,  a  current  flowing  away 
from  the  reader  is  generally  indicated  by  a  cross  (represent- 
ing the  tail  of  an  arrow)  in  the  small  circle  representing  the 
cross-section;  a  current  flowing  toward  the  reader  is  indi- 
cated by  a  dot  (representing  the  point  of  an  arrow)  in  the 
circle,  as  shown  in  Fig.  7c. 


rr 

i 

t 

s— 

1 

7N. 

• 

v  -^ 

i 

t 

"0 

W 

(a) 


FIG.  7. 


17.  The  Solenoid. — When  a  wire  is  wound  in  the  form 
of  a  coil,  and  current  is  passed  through  it,  it  exhibits  all  of 
the  properties  of  polarity,  attraction  and  repulsion,  that  are 
shown  by  a  magnet,  but  the  magnetic  field  disappears  when 
the  current  ceases  to  flow.  Such  a  coil  is  called  a  solenoid. 
If  it  be  suspended  at  the  center  so  as  to  be  able  to  move  freely 
with  its  axis  horizontal,  it  will  assume  a  north  and  south 
direction.  If  a  pole  of  a  magnet  be  brought  near  it,  one  end 
of  the  coil  will  be  attracted  and  the  other  repelled.  If  the 
end  of  a  bar  of  soft  iron  be  brought  near  one  end  of  the 
solenoid,  the  bar  will  be  magnetized  by  induction  and 
drawn  into  the  solenoid  in  accord  with  the  principle  stated 
in  Article  13.  If  the  axes  of  coil  and  bar  are  horizontal 


20 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


and  there  be  no  mechanical  friction  where  the  bar  enters 
the  coil,  the  center  of  the  bar  will  go  to  the  center  of  the 
coil,  because  this  is  the  only  position  where  the  forces  acting 
between  bar  and  coil  are  in  equilibrium.  If  the  axes  of  coil 
and  bar  are  vertical,  the  bar  will  be  pulled  up  until  the  dif- 
ference between  the  attractive  forces  just  balances  the  force 
of  gravity,  and  will  be  suspended  at  that  point. 

Fig.  8  illustrates  the  field  associated  with  a  solenoid.  In 
accord  with  the  rule  given  in  the  last  article,  the  following 
rule  may  be  used  for  determining  the  direction  of  the  flux  in  a 
solenoid:  If  one  looks  along  the  axis  of  a  coil  and  the  cur- 


V-X          |    V_X  ^-/  V_X  \^       |        \^      ^       \_X       i        \_X  V 

/     i    \  )  '      ^ 


FIG.  8. 


rent  is  flowing  in  a  clockwise  direction  around  the  coil,  the 
direction  of  the  flux  inside  the  coil  will  be  away  from  the 
observer. 

18.  Action  of  a  Magnetic  Field  on  a  Wire  Carrying  Cur- 
rent.— If  a  wire  carrying  current  is  placed  across  a  magnetic 
field  not  its  own,  a  force  will  be  exerted  upon  the  wire. 
Suppose  the  lines  of  force  of  the  field  in  which  the  wire  is 
placed  pass  from  left  to  right  as  shown  in  Fig.  9,  while  the 
lines  of  force  due  to  the  wire  pass  clockwise  around  the  wire. 
The  result  is  that  the  intensity  of  the  field  is  increased  above 
the  wire,  and  weakened  below  the  wire,  and  a  downward 
force  will  be  exerted  on  the  wire.  This  may  be  looked  upon 


ELECTRICITY  AND  MAGNETISM 


21 


as  the  result  of  the  natural  tension  in  the  lines  of  force  and 
their  tendency  to  shorten  themselves.  The  direction  of 
the  force  is  always  at  right  angles  both  to  the  field  and  to  the 
wire  and  toward  that  side  of  the  wire  where  the  field  is 
weakened.  The  value  of  this  force  (per  unit  length  of  wire) 
is  greatest  when  the  wire  is  at  right  angles  to  the  field.  A 
rule  for  determining  the  direction  of  the  force  in  such  a  case 
is  known  as  the  "  left-hand  rule."  If  the  forefinger  of  the 
left  hand  be  pointed  in  the  direction  of  the  main  field  and 
the  middle  finger  placed  at  right  angles  to  the  forefinger 
and  pointed  in  the  direction  of  the  current,  then  the  thumb, 
placed  at  right  angles  to  both  fore  and  middle  fingers,  will 
point  in  the  direction  of  the  force  action  upon  the  wire. 


FIG.  9. 


19.  Unit  Current. — The  force  action  described  above 
varies  with  the  strength  of  the  current,  with  the  intensity  of 
the  magnetic  field,  with  the  length  of  the  wire  immersed 
in  the  field,  and  with  the  angle  which  the  wire  makes  with 
the  direction  of  the  field.  When  the  field  and  the  wire  are 
at  right  angles  to  each  other  the  force  is  proportional  to 
the  product  of  the  current,  the  field  intensity,  and  the  length 
of  the  wire.  This  fact  is  made  the  basis  of  the  fundamental 
unit  of  current.  This  fundamental  unit,  known  as  the  c.g.s. 
unit,  or  abampere,  is  a  current  of  such  strength  that  the  wire 
in  which  it  flows  has  exerted  upon  it  a  force  of  one  dyne  per 
centimeter  of  length  when  the  wire  is  at  right  angles  to  a  mag- 
netic field  of  unit  intensity.  This  is  considered  to  be  too 
large  for  practical  purposes,  and  a  unit  one-tenth  as  large 
and  called  an  ampere  is  used  as  the  practical  unit. 

If  the  intensity  of  the  field  is  H  lines  per  square  centi- 


22  ELECTRIC  AND  MAGNETIC  CIRCUITS 

meter,  the  strength  of  the  current  /  amperes,  and  the 
length  of  the  wire  I  centimeters,  then  the  force  on  the  wire  in 
dynes  is 

'-#• 

When  the  wire  is  parallel  with  the  field  no  force  acts  on 
the  wire.  When  the  angle  between  the  wire  and  the  direc- 
tion of  the  field  is  6,  then  the  field  may  be  thought  of  as  having 
two  components,  one,  H  cos  0,  parallel  with  the  wire,  and 
the  other,  H  sin  6,  at  right  angles  to  the  wire.  The  former 
component  has  no  action  on  the  wire,  while  the  latter  pro- 
duces a  force, 

f-flT 
F  =  ~sme.  (2) 

The  exact  determination  of  the  value  of  a  current  by 
measuring  this  force  action  is  an  extremely  difficult  process. 
However,  it  has  been  done  and  the  value  of  an  ampere  cal- 
culated in  terms  of  an  easier  method  of  measurement,  as 
explained  in  the  next  paragraph. 

20.  International  Unit  of  Current. — It  has  been  discov- 
ered that  if  an  electric  current  be  passed  through  certain 
chemical  solutions,  the  substance  will  be  decomposed.  For 
example,  if  a  current  flows  into  a  solution  of  copper  sulphate 
by  means  of  a  rod  or  plate  of  solid  conducting  material,  and 
out  of  it  through  another  rod  or  plate,  the  copper  will  be 
deposited  out  of  the  solution;  and  if  the  rod  by  means  of 
which  the  current  leaves  the  solution  is  properly  selected  and 
prepared,  the  copper  will  adhere  to  it.  This  is  called 
electrolytic  deposition.  The  amount  of  material  deposited 
in  this  way  is  found  to  be  proportional  to  the  quantity  of 
electricity  which  passes  through  the  solution;  that  is,  to 
the  product  of  the  strength  of  the  current  which  flows,  and 
the  tune  during  which  it  flows.  The  electrolytic  deposition 
of  silver  out  of  a  solution  of  silver  nitrate  on  to  a  plate  of  pure 
silver  is  found  to  be  a  most  accurate  way  of  measuring  an 
electric  current.  It  is  found  that  1  ampere  measured  by 


ELECTRICITY  AND  MAGNETISM  23 

the  fundamental  method  mentioned  in  the  last  paragraph 
will  under  certain  carefully  observed  specifications  deposit 
1.118  milligrams  of  silver  per  second.  For  these  specifi- 
cations, see  Circular  No.  60,  U.  S.  Bureau  of  Standards. 
The  ampere,  thus  measured,  was  made  the  legal  unit  of  cur- 
rent by  Act  of  Congress  in  1894,  and  was  adopted  inter- 
nationally as  the  standard  method  of  determining  the  unit 
of  current  by  the  International  Electric  Congress  at  Lon- 
don in  1908;  the  ampere  measured  in  this  way  is  called  the 
International  Unit  of  Current,  or  International  Ampere. 

21.  The  Coulomb. — The  coulomb  has  been  mentioned  as 
the  unit  quantity  of  electricity.     Since  the  ampere,  or  the 
unit  rate  of  flow  of  electricity,  is  adopted  as  the  primary 
unit,  the  value  of  the  coulomb  must  be  defined  in  terms  of 
the  ampere.    Unit  quantity  of  electricity,  or  the  coulomb, 
is  that  quantity  which  flows  past  a  given  point  in  one  second 
when  the  rate  of  flow  is   1   ampere.     When  the  current 
flowing  in  a  circuit  is  8  amperes,  for  instance,  the  quantity 
of  electricity  passing  through  the  circuit  is  8  coulombs  per 
second.     If  8  amperes  flows  through  a  circuit  for  twenty 
seconds,  the  total  quantity  of  electricity  passing  through 
the  circuit  during  that  time  is  160  coulombs.     If  1  ampere 
flows  for  one  hour,  3600  coulombs  would  pass  through  the 
circuit.     In  practical  work  the  ampere-hour,  equal  to  3600 
coulombs,  is  generally  used  as  the  unit  of  quantity.     Thus, 
15  amperes  flowing  for  six  hours  would  give  90  ampere- 
hours  and  would  be   called   90   ampere-hours  instead  of 
324,000  coulombs. 

22.  The  Galvanometer. — An  instrument  commonly  used 
to  indicate  the  flow  of  electricity  is  the  galvanometer.     It 
may  also  be  used  within  certain  limits  to  measure  the  value 
of  a  current  or  of  the  quantity  of  electricity  passed  through 
it  in  a  very  short  interval  of  time.     The  principle  on  which 
such  instruments  operate  is  the  force  action  between  a  mag- 
netic field  and  a  wire  carrying  an  electric  current.     There 
are  two  general  methods  of  construction:   One,  in  which  a 
small  permanent  magnet  is  suspended  within  a  coil  of  wire, 


24  ELECTRIC  AND  MAGNETIC  CIRCUITS 

at  right  angles  to  the  axis  of  the  coil;  and  the  other,  in 
which  a  light  coil  of  wire  is  suspended  between  the  poles  of  a 
magnet  with  its  axis  at  right  angles  to  the  direction  of  field 
due  to  the  magnet.  The  movement  of  the  suspended  part 
may  be  indicated  by  a  light  pointer  attached  to  it,  or  by 
means  of  a  small  mirror  attached  to  it  and  from  which  a  beam 
of  light  is  reflected  to  a  scale,  or  from  which  a  scale  is  ob- 
served from  a  fixed  position.  In  the  case  of  a  moving 
magnet  the  zero  position  is  fixed  by  another  magnet  placed 
near  the  instrument  or  by  the  earth's  magnetic  field.  The 
coil  and  the  fixed  field  are  so  placed  with  respect  to  each 
other  that  the  suspended  magnet  is  held  in  the  plane  of  the 
coil  when  in  the  zero  position,  that  is,  with  no  current  in 
the  coil.  When  current  flows  through  the  coil  the  needle 
takes  up  a  position  depending  on  a  relative  strength  of  the 
fixed  field  which  tends  to  hold  it  in  the  zero  position  and  the 
field  produced  by  current  in  the  coil  which  tends  to  turn  it 
to  a  position  parallel  with  its  axis;  that  is,  perpendicular 
to  its  plane.  In  the  case  of  the  suspended  coil,  the  zero 
position  is  fixed  by  springs  attached  to  it,  and  the  current 
is  led  into  and  out  of  the  coil  through  these  springs.  When 
current  flows  in  the  coil,  it  takes  up  a  position  depending 
on  the  relative  strength  of  the  spring  and  the  combined 
strength  of  the  fields  which  tends  to  turn  the  coil  into  a  posi- 
tion with  its  axis  parallel  to  the  fixed  field.  A  more  com- 
plete description  of  these  instruments  may  be  found  in  any 
good  hand-book  for  electrical  engineers. 

Within  quite  narrow  limits  the  deflection  of  a  galvanom- 
eter needle  or  coil  is  proportional  to  the  current  flowing,  but 
such  instruments  are  more  commonly  used  for  determining 
the  condition  of  zero  potential  difference  between  two  points. 
They  may  be  constructed  so  as  to  indicate  extremely  small 
currents  (as  small  as  10  ~12  amperes)  and  therefore  for  all 
practical  purposes  when  no  deflection  is  discernible  there  is 
no  potential  difference  between  the  pomts  where  the  instru- 
ment is  connected.  When  the  inertia  of  the  moving  part 
is  sufficient  to  prevent  its  getting  into  motion  until  a  momen- 


ELECTRICITY  AND  MAGNETISM  25 

tary  flow  of  electricity  has  passed  through  the  coil,  the 
swing  which  then  takes  place  is,  within  narrow  limits,  pro- 
portional to  the  amount  of  electricity  which  has  passed 
through.  Such  a  galvanometer  is  called  ballistic  and  is 
frequently  used  for  measuring  small  quantities  of  electricity. 
Whenever  a  galvanometer  is  used  to  measure  a  current  or  a 
quantity  of  electricity,  it  must  first  be  calibrated  by  reading 
its  deflections  when  known  current  or  quantities  are  passed 
through  it.  When  used  for  determining  the  ratio  of  two 
currents  or  two  quantities,  this  calibration  is  unnecessary. 

The  type  of  galvanometer  using  a  movable  coil  sus- 
pended between  permanent  magnets  is  known  as  the 
D' Arson val  Galvanometer.  The  principle  of  this  galva- 
nometer is  the  most  common  one  used  in  the  construction  of 
direct  current  ammeters  and  voltmeters  for  measuring  cur- 
rent and  voltage.  See  Figs.  28  and  29.  In  such  meters, 
however,  the  coil  is  pivoted  between  jeweled  bearings, 
and  held  in  its  zero  position  by  spiral  springs.  A  pointer 
is  rigidly  attached  to  the  coil  and  swings  over  a  graduated 
scale. 


CHAPTER   III 
ELECTRIC  CIRCUITS 

23.  Resistance. — Mention  has  been  made  of  the  fact 
that  when  an  electric  current  flows  in  a  wire,  heat  is  generated 
therein.  This  fact  leads  immediately  to  the  conclusion  that 
the  wire  must  possess  some  property  by  which  it  opposes  the 
flow  of  current;  if  the  wire  offered  no  opposition,  no  effort 
would  be  required  to  send  the  current  through  the  wire,  no 
work  would  be  done,  and  no  heat  generated.  It  has  been 
proven  experimentally  that  all  materials  possess  an  inherent 
property  by  which  they  oppose  the  flow  of  electricity — some 
offering  very  great  opposition,  some  comparatively  little. 
This  property  of  opposition  is  called  resistance.  Those 
materials  which  offer  comparatively  small  resistance  are 
called  conductors;  the  principal  materials  in  this  class  are 
the  metals,  carbon,  and  solutions  of  mineral  salts  and  acids. 
The  materials  which  offer  great  opposition  are  called  insu- 
lators; glass,  porcelain,  mica,  rubber,  silk,  paper,  paraffine, 
shellac  and  oil  are  examples  of  good  insulators. 

The  energy  consumed  hi  overcoming  resistance  is  trans- 
formed into  heat.  The  rate  at  which  heat  is  generated  in  a 
wire  has  been  proven  to  be  proportional  to  the  square  of 
the  current  flowing  in  the  wire;  that  is,  equal  to  a  constant 
tunes  the  square  of  the  current.  This  is  known  as  Joule's 
Law.  The  value  of  the  constant  is  called  the  resistance  of 
the  wire.  The  fundamental,  or  c.g.s.  unit  of  resistance  is 
that  resistance  in  which  1  abampere  of  current  will  generate 
heat  energy  at  the  rate  of  1  erg  per  second;  it  is  called  an 
abohm.  The  practical  unit  of  resistance  is  that  resistance  in 
which  a  current  of  1  ampere  will  generate  1  joule  of  heat  energy 
in  one  second.  This  unit  is  called  an  ohm,  and  is  equal  to 

26 


ELECTRIC  CIRCUITS  27 

10°  abohms.  Since  the  rate  at  which  energy  is  transformed, 
or  work  is  done,  is  power,  and  since  work  done  at  the  rate 
of  1  joule  per  second  is  1  watt,  an  ohm  may  be  defined  as 
that  resistance  in  which  1  ampere  develops  power  at  the 
rate  of  1  watt.  If  the  power  developed  in  a  wire  is  64  watts, 
when  a  current  of  4  amperes  is  flowing  in  it,  the  resistance 
of  the  wire  is  64/16=4  ohms.  That  is,  the  resistance  of 
the  wire  is  equal  to  the  power  developed  in  it  divided  by  the 
the  square  of  the  current  flowing;  or,  the  current  flowing 
in  a  wire  is  equal  to  the  square  root  of  the  quotient  obtained 
by  dividing  the  power  by  the  resistance.  Put  into  form  of 
an  equation,  the  law  is 

P=RP,  (3) 

where  P  is  the  power  in  watts  lost  in  heating  a  conductor 
of  resistance  R  when  a  current  /  flows  through  it. 

In  comparing  resistances,  it  is  necessary,  of  course,  to 
have  a  standard.  By  careful  measurement,  it  has  been 
found  that  a  column  of  pure  mercury  measuring  106.3 
cm.  long  at  the  temperature  of  melting  ice,  of  uniform 
cross-sectional  area,  and  weighing  14.4521  gm.,  has  a  resist- 
ance of  1  ohm  as  defined  above.  This  is  known  as  the  Inter- 
national Ohm,  and  secondary  standards  are  made  by 
comparison  with  it. 

24.  Ohm's  Law  and  Electromotive  Force. — It  has  been 
proven  by  experiment  that  the  current  which  flows  in  a  con- 
ductor is  directly  proportional  to  the  electromotive  force 
which  is  applied  to  that  conductor,  and  inversely  propor- 
tional to  the  resistance  of  the  conductor.  This  is  known 
as  Ohm's  Law,  and  is  the  electrical  application  of  the 
general  law  that  the  magnitude  of  any  effect  varies  directly 
with  the  magnitude  of  the  cause  and  inversely  with  the 
magnitude  of  the  opposition.  We  have  defined  the  prac- 
tical units  of  current  and  of  resistance  and  we  can  now  define 
the  practical  unit  of  electromotive  force  as  that  electromotive 
force  which  will  cause  a  current  of  1  ampere  to  flow  through  a 
resistance  of  1  ohm.  As  already  stated,  this  unit  is  called 


28  ELECTRIC  AND  MAGNETIC  CIRCUITS 

a  volt.  The  absolute  unit,  or  abvolt,  is  that  e.m.f.  which 
will  cause  1  abampere  of  current  to  flow  in  1  abohm  of 
resistance;  the  practical  unit,  or  volt,  is  equal  to  108  abvolts. 
Ohm's  Law  is  one  of  the  most  important  laws  in  Electrical 
Engineering  and  careful  attention  must  be  given  to  its 
operation.  Expressed  in  the  form  of  an  equation  the  law  is, 

7=|,     or    E  =  RI,    or    R~t  (4) 

where  I  is  the  current  which  an  electromotive  force  E  will 
cause  to  flow  in  a  resistance  R.  This  law  also  means  that 
when  a  current  is  flowing  through  a  given  wire,  the  differ- 
ence of  potential  between  any  two  points  is  equal  to  the 
product  of  the  current  flowing  and  the  resistance  of  the  por- 
tion of  the  wire  between  the  two  points.  An  extended  dis- 
cussion of  the  application  of  Ohm's  Law  will  be  given  in 
later  articles. 

When  a  current  flows  through  a  wire  the  difference  of 
potential  between  two  points  is  sometimes  called  the  poten- 
tial drop  or  the  drop  in  potential  and  is  often  abbreviated 
as  p.d.  When  the  word  "  drop  "  is  used  it  must  be  under- 
stood that  it  occurs  in  the  direction  of  current  flow,  just  as  a 
drop  in  head  or  a  drop  in  pressure  in  a  pipe  line  carrying 
water  means  that  the  pressure  is  less  in  the  direction  of 
water  flow.  If  the  direction  opposite  to  the  flow  is  con- 
sidered, there  would  be  a  rise  of  potential.  The  term 
"  rise  of  potential  "  is  also  sometimes  used  to  denote  the 
increase  in  potential  from  one  terminal  to  the  other  inside 
of  a  source  of  electromotive  force.  This  rise  of  potential  is 
caused  by  the  unknown  process  which  generates  electro- 
motive force  and  is  in  the  direction  of  current  flow  since 
current  is  assumed  to  flow  out  from  a  source  of  e.m.f.  by 
way  of  the  terminal  having  the  higher  potential  and  in  at 
the  terminal  having  the  lower  potential.  It  should  be  clearly 
understood  that  potential  refers  to  the  condition  at  one 
point  and  that  difference  of  potential,  drop  in  potential 


ELECTRIC  CIRCUITS  29 

or  rise  in  potential  refers  to  the  difference  in  conditions 
between  two  points. 

25.  Chemical  Sources  of  E.M.F. — While  it  is  not  within 
the  scope  of  this  text  to  enter  into  a  discussion  of  details  in 
regard  to  the  theory  of  generation  of  electromotive  force, 
there  are  certain  principles  relating  to  the  action  of  sources 
of  electromotive  force  which  should  be  learned  at  this  time. 

When  two  plates  or  rods  of  different  metals,  or  a  piece  of 
metal  and  a  piece  of  carbon  are  placed  in  certain  chemical 
solutions,  an  electromotive  force  is  generated  which  estab- 
lishes a  difference  of  potential  between  the  two  plates. 
Such  an  arrangement  is  called  a  voltaic  cell.  The  solution 
used  in  a  voltaic  cell  is  called  the  electrolyte  and  the  two 
conducting  rods  or  plates  which  are  placed  in  it  are  called 
electrodes.  The  value  of  the  electromotive  force  produced 
by  a  voltaic  cell  varies  with  different  materials  used.  For 
example,  copper  and  zinc  in  a  solution  of  zinc  sulphate 
develops  an  e.m.f.  of  about  1  volt;  carbon  and  zinc  in  a  solu- 
tion of  ammonium  chloride  develops  an  e.m.f.  of  about  1.4 
volts;  carbon  and  zinc  in  a  solution  of  dilute  sulphuric  acid 
develops  about  2  volts.  The  direction  in  which  current 
will  flow  from  a  voltaic  cell  depends  also  upon  the  materials 
used  for  the  electrodes.  When  copper  and  zinc  are  used,  the 
current  will  flow  out  at  the  copper  terminal;  when  carbon 
and  zinc  are  used,  it  will  flow  out  at  the  carbon  terminal; 
that  is,  in  the  first  case,  the  copper  terminal  is  at  a  higher 
potential  than  the  zinc,  and  in  the  second  case  the  carbon 
terminal  is  at  a  higher  potential  than  the  zinc.  The  elec- 
trode at  which  the  current  flows  out  of  a  cell  is  called  the 
positive  electrode  and  the  one  at  which  the  current  flows 
into  the  cell  is  called  the  negative  electrode. 

In  every  voltaic  cell  when  current  is  allowed  to  flow 
from  it,  the  chemical  composition  of  the  electrolyte,  or  of  the 
electrodes,  or  both,  undergoes  a  change  as  the  chemical 
energy  gradually  becomes  exhausted.  If  a  cell  is  con- 
nected to  another  source  of  electromotive  force  and  current 
be  sent  through  it  in  a  direction  opposite  to  that  of  its  own 


30  ELECTRIC  AND  MAGNETIC  CIRCUITS 

e.m.f.  the  chemical  action  will  be  reversed  and  there  will 
be  a  tendency  to  restore  the  cell  to  its  original  condition; 
but  in  most  cases  it  is  found  to  be  impracticable  by  this 
means  to  restore  the  chemical  energy  in  any  considerable 
amount,  owing  to  the  fact  that  in  the  original  operation  of 
the  cell,  there  are  certain  local  chemical  actions  which  are 
not  reversible,  and  owing  to  the  loss  of  certain  constituents 
(principally  gases)  out  of  the  electrolyte.  There  are,  how- 
ever, a  few  combinations  into  which  from  75  per  cent  to 
90  per  cent  of  the  energy  can  be  restored  by  reversing  the 
operation.  These  combinations  are  known  as  storage  cells. 
The  two  most  important  of  these  are  the  lead  cell,  and  the 
alkaline  or  Edison  cell.  The  lead  cell  is  made  by  filling 
two  lead  grids  with  a  paste  of  lead  sulphate  and  placing 
them  in  an  electrolyte  of  dilute  sulphuric  acid.  When  a 
current  is  sent  through  the  cell  (which  process  is  known 
as  charging)  the  lead  sulphate  on  the  plate  where  the 
current  enters  is  changed  to  lead  peroxide  and  that  on 
the  plate  where  the  current  leaves  is  reduced  to  spongy  lead. 
When  all  or  most  of  the  lead  sulphate  has  been  changed  in 
this  way  to  lead  peroxide  and  spongy  lead,  the  cell  is  said 
to  be  charged  and  may  be  used  as  an  ordinary  voltaic  cell 
and  will  produce  current  until  most  of  the  lead  and  lead 
peroxide  is  changed  back  to  lead  sulphate.  When  the  cell 
is  charged  the  plate  containing  the  lead  peroxide  is  positive, 
or  at  the  higher  potential,  and  the  one  containing  the  spongy 
lead  is  negative,  or  at  the  lower  potential.  The  e.m.f. 
of  the  cell  is  about  2.2  volts  when  fully  charged  and  decreases 
at  first  slowly  and  then  rapidly  to  zero  as  it  is  discharged 
by  using  it  as  a  source  of  energy.  If  the  discharge  is  con- 
tinued beyond  the  point  where  the  p.d.  at  the  terminals 
is  about  1.8  volts,  the  reversibility  of  the  cell  is  greatly 
impaired,  and  for  practical  purposes  a  cell  is  considered 
to  be  discharged  when  its  terminal  p.d.  has  dropped  to 
1.8  volts. 

When  a  cell  is  being  charged  the  specific  gravity  of  the 
electrolyte  increases,  and  when  fully  charged  it  varies  from 


ELECTRIC  CIRCUITS  31 

about  1.2  to  1.3,  depending  on  the  service  for  which  it  is 
intended.  When  discharged  the  specific  gravity  will  gen- 
erally be  from  1.13  to  1.18.  The  specific  gravity  is  a  better 
criterion  as  to  the  condition  of  a  cell  than  is  the  e.m.f. 
The  open-circuit  voltage  affords  little,  if  any,  indication  as 
to  the  condition  of  a  cell,  until  it  is  completely  discharged. 

The  Edison  cell  consists  of  nickel  peroxide  for  the  posi- 
tive plate  and  finely  divided  iron  in  a  suitable  container 
for  the  negative  plate,  with  a  solution  of  potassium  hydrate 
as  the  electrolyte.-  This  cell  gives  an  initial  open-circuit 
voltage  of  about  1.5  when  charged,  which  falls  to  about 
1.4  on  closed  circuit  and  gradually  drops  to  about  1  volt 
when  discharged.  Results  seem  to  show  that  this  cell  is 
much  less  liable  to  injury  from  overcharge  or  over-discharge 
than  is  the  lead  type. 

26.  Voltage  Relations  in  Battery  Circuits. — It  must  be 
understood  that  e.m.f.  is  generated  in  a  cell  whether  or  not 
the  terminals  are  connected  to  a  circuit.  However,  when 
the  circuit  is  closed  and  current  allowed  to  flow,  the  total 
e.m.f.  developed  in  the  cell  becomes  less  as  time  goes  on,  on 
account  of  certain  chemical  changes  which  take  place  as  the 
chemical  energy  of  the  cell  is  transformed  into  electrical 
energy.  The  complete  circuit  includes  the  solution  within 
the  cell  which  is,  of  course,  a  conductor  and  has  resistance. 
The  current  which  flows  at  any  time  is  by  Ohm's  Law  equal 
to  the  total  e.m.f.  divided  by  the  total  resistance  of  the  cir- 
cuit; a  portion  of  the  total  e.m.f.  is  used  in  overcoming 
the  resistance  inside  the  cell,  while  the  rest  is  used  in  over- 
coming the  resistance  connected  between  the  terminals 
externally.  The  amount  of  e.m.f.  used  in  overcoming  the 
resistance  of  the  cell  is,  again  by  Ohm's  Law,  equal  to  the 
product  of  the  current  flowing  and  the  resistance  of  the  cell, 
and  the  amount  used  on  the  outside  resistance  is  equal  to 
the  product  of  the  current  and  the  outside  resistance;  that 
is,  the  potential  difference  at  the  terminals  is  less  than  the 
total  e.m.f.  of  the  cell  by  the  amount  of  e.m.f.  used  in  over- 
coming the  internal  resistance  of  the  cell.  This  is  an 


32  ELECTRIC  AND  MAGNETIC  CIRCUITS 

important  principle  and  must  be  thoroughly  mastered.  If 
a  cell  has  an  e.m.f.  of  1.4  volts  and  an  internal  resistance  of 
3  ohms,  and  is  connected  to  an  external  resistance  of  4  ohms, 
see  Fig.  10,  the  total  resistance  of  the  circuit  will  be  3  +4=7 
ohms  and  the  current  will  be  1.4/7=0.2  ampere.  The 
portion  of  the  e.m.f.  which  is  used  inside  the  cell  to  overcome 
its  resistance  is  therefore,  by  Ohm's  Law,  3x0.2=0.6  volt. 
The  potential  difference  at  the  terminals  is  therefore  1.4  — 
0.6=0.8  volt. 

Stated  in  a  different  way,  we  may  say  that  there  is  a  rise 
of  potential  of  1.4  volts  from  the  negative  terminal,  a, 
through  the  cell  to  the  positive  terminal  6,  due  to  the  e.m.f. 
of  the  cell,  but  when  a  current  of  0.2  ampere  flows,  there  is  a 
resistance  drop  of  potential  of  0.6  volt  within  the  cell,  leaving 


.Rx=4  ohms 


a 

i 
< 

E  =  1.4  volts 

—                                                                                                                     4 

b 

7?^=3.0  ohms              ; 

< 
< 

FIG.  10. 


a  net  rise  of  potential  of  0.8  volt.  With  reference  to  the 
external  circuit,  this  0.8  volt  is  a  fall  of  potential.  That 
portion  of  the  e.m.f.  which  is  used  to  overcome  internal 
resistance  is  commonly  called  the  internal  resistance  drop, 
meaning  that  the  value  of  the  potential  difference  at  the 
terminals  is  less  than  the  value  of  the  e.m.f.  of  the  cell  by 
an  amount  equal  to  the  product  of  the  current  and  the 
internal  resistance.  Since  R  is  the  symbol  always  used  for 
resistance,  and  /  the  symbol  used  for  current,  this  drop  is 
also  very  commonly  called  the  internal  RI  drop.  The 
expression  RI  drop,  is  also  applied  to  the  difference  of  poten- 
tial between  any  two  points  in  a  wire  when  a  current  is 
flowing  through  it.  When  no  current  flows,  the  RI  drop  is 
zero,  and  the  potential  difference  at  the  terminals  is  equal 
to  the  e.m.f.  of  the  cell. 

The  word  "  voltage  "  is  very  commonly  used  to  express 


ELECTRIC  CIRCUITS 


33 


both  e.m.f.  and  potential  difference;  thus,  when  the  e.m.f. 
of  a  cell  is  1.4,  it  might  be  said  that  its  voltage  is  1.4,  and 
when  the  potential  difference  between  its  terminals  is  0.8, 
it  might  be  said  that  the  voltage  at  its  terminals  is  0.8. 

27.  Cells  in  Series. — When  two  or  more  cells  are  con- 
nected together  in  order  to  obtain  more  power,  the  com- 
bination is  called  a  "  battery. "  Sometimes  the  word  bat- 
tery is  used  to  indicate  a  single  cell,  but  this  is  an  incorrect 
use  of  the  word.  When  a  number  of  cells  are  connected 
so  that  the  same  current  passes  through  each  of  them  one 
after  another,  they  are  said  to  be  in  series.  If  one  terminal 
of  each  cell  is  connected  to  a  common  point,  and  the  other 


HHH* 


B 


/vwvwww 


HHH 


R 


FIG.  11. 


FIG.  12. 


terminal  of  each  cell  connected  to  another  common  point, 
they  are  said  to  be  in  parallel.  When  connected  in  series, 
with  the  negative  terminal  of  each  cell  connected  to  the 
positive  terminal  of  its  neighbor,  the  total  e.m.f.  of  the  bat- 
tery is  the  sum  of  the  e.m.f 's  of  the  individual  cells.  A 
common  method  of  representing  a  cell  is  to  draw  two  par- 
allel lines  as  shown  by  the  vertical  lines  in  Fig.  1 1  where  the 
longer  thin  line  represents  the  positive  terminal,  and  the 
shorter  thick  line  represents  the  negative  terminal.  Fig. 
11  represents  a  battery  of  four  cells  connected  in  series  with 
all  cells  connected  so  that  their  e.m.f  s  act  in  the  same  way 
around  the  circuit.  R  represents  a  resistance  connected 
to  the  terminals  A  and  B  of  the  battery.  Suppose  that 
each  cell  has  an  e.m.f.  of  1.4  volts  and  an  internal  resistance 
of  3  ohms,  while  the  resistance  R  is  16  ohms.  The  e.m.f. 


34 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


of  the  battery  is  4x1.4=5.6  volts  and  the  total  resistance 
of  the  battery  is  4  X3  =  12  ohms;  therefore  the  total  resist- 
ance of  the  circuit  is  12  +  16  =28  ohms,  and  the  current  will 
be  5.6/28=0.2  ampere.  The  e.m.f.  used  in  overcoming 
the  battery  resistance  is  0.2x12=2.4  volts,  while  that 
used  in  overcoming  the  external  resistance,  R,  is  0.2  X 16  =  3.2 
volts.  The  last  result  is,  of  course,  the  potential  difference 
at  the  terminals  of  the  battery  and  may  also  be  calculated 
by  subtracting  2.4  from  5.6.  In  Fig.  12  is  shown  a  battery 
of  four  cells  with  one  cell  (the  right-hand  one)  reversed, 
that  is,  it  is  so  connected  that  its  e.m.f.  acts  in  a  direction 
opposite  to  that  of  the  other  three.  In  this  case,  if  each  cell 
has  an  e.m.f.  of  1.4  volts,  the  total  e.m.f.  acting  on  the  cir- 
cuit is  (3x1 .4)  - 1 .4  =  4.2  - 1 .4  =  2.8  volts.  The  two  right- 
hand  cells  balance  each  other,  and  add  nothing  to  the 
e.m.f.  of  the  circuit.  Note,  however,  that  the  internal 
resistance  of  a  cell  opposes  the  flow  of  current  through  the 
cell,  no  matter  which  way  the  current  flows. 

28.  Cells  in  Parallel. — In  any  cell  there  is  a  limit  to  the 
current  which  can  be  allowed  to  flow  through  it,  without 


/WWWWWVWN 


FIG.  13. 


serious  reduction  of  its  e.m.f.  and  rapid  deterioration  from 
too  rapid  chemical  action.  When  more  current  is  desired 
than  one  cell  can  stand,  it  is  common  practice  to  connect 
them  in  parallel.  Fig.  13  shows  a  battery  of  4  cells  in  par- 
rallel  connected  to  a  resistance  R.  In  this  method  of  con- 


ELECTRIC  CIRCUITS 


35 


nection  the  total  e.m.f.  is  only  that  of  one  cell,  but  the  total 
current  divides  between  the  cells,  each  cell  carrying,  in  the 
circuit  shown,  one-fourth  of  the  current.  If  a  battery  be 
connected,  as  in  Fig.  14,  the  total  allowable  current  will  be 


A/VWWWWWV\ 

FIG.  14. 

three  times  that  for  one  cell,  and  the  total  e.m.f.  will  be 
twice  that  for  one  cell. 

It  is  sometimes  a  matter  of  difficulty  to  understand  why, 
when  two  or  more  cells  are  connected  in  parallel,  the  total 
e.m.f.  acting  on  the  circuit  -is  equal  only  to  the  e.m.f.  of  one 
cell.  Consider  Fig.  15,  which  shows  two  cells  in  parallel 
between  the  points  A  and  B. 
Evidently  no  current  can  flow 
around  this  circuit  because  the 
two  cells  act  in  opposite  direc- 
tions around  the  circuit  and 
their  e.m.f 's  are  supposed  to 
be  equal.  By  Ohm's  Law, 
the  difference  of  potential 
between  any  two  points  along  a  given  wire  is  equal  to  RI, 
where  R  is  the  resistance  between  the  points,  and  7  is  the 
current  flowing.  Therefore,  in  this  case,  there  is  no  differ- 
ence of  potential  between  the  two  positive  terminals;  that 
is,  the  two  positive  terminals  and  the  point  B  are  at  the 
same  potential.  Similarly,  the  two  negative  terminals  and 
the  point  A  are  at  the  same  potential.  Therefore,  the 
difference  of  potential  between  A  and  B,  considered  either 


FIG.  15. 


36 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


through  the  upper  cell  or  through  the  lower  cell,  is  equal  to 
the  e.m.f.  of  that  cell. 

Let  A  and  B  be  connected  through  an  external  resistance 
R  as  in  Fig.  16;  consider  that  the  two  wires  leading  from  A 
to  the  negative  terminals  of  the  cells  have  the  same  resistance 
and  likewise  the  two  wires  leading  from  the  two  positive 
terminals  to  B.  If  the  two  cells  have  the  same  e.m.f. 
and  the  same  internal  resistance,  the  total  current  will 
divide  equally  between  them.  The  drop  in  potential  from 
each  positive  terminal  to  the  point  B  is  the  same,  and  like- 


/WWWWVWWN 


FIG.  16. 

wise  the  drop  from  A  to  each  negative  terminal.  There- 
fore, the  potential  difference  between  A  and  B  is  equal  to 
the  e.m.f.  of  either  cell  minus  the  drop  in  the  wires  leading 
to  and  from  the  cell  and  minus  the  drop  within  the  cell. 
The  p.d.  between  A  and  B  is  also  equal  to  the  product  of 
the  external  resistance  R  and  the  current  7. 

29.  Power  and  Energy  in  an  Electric  Circuit. — From 
Joule's  Law  we  have  the  relation  that  the  power  developed 
in  a  resistance  R  due  to  a  current  I  is  equal  to  RI2',  that  is, 

P  =  RP  (5) 

We  also  have  from  Ohm's  Law  that 


(6) 


ELECTRIC  CIRCUITS  37 

Substituting  (6)  into  (5),  we  get, 

P  =  EI  (7) 

which  shows  that  the  power  in  watts  developed  in  an  electric 
circuit  is  equal  to  the  product  of  the  current  and  the  e.m.f . 
This  equation  must  be  understood  to  give  the  power  only  be- 
tween the  points  having  a  potential  difference  E.  The  equa- 
tion is  true  whether  the  voltage  E  is  all  used  in  overcoming 
resistance  or  partly  used  in  overcoming  a  counter  or  back- 
e.m.f . ;  for  in  the  latter  case,  the  back  e.m.f.  can  be  replaced 
by  a  resistance  r  which  will  hold  the  current  down  to  the  same 
value  that  it  has  when  the  back  e.m.f.  is  in  circuit  and  the 


vwvwvwv 

0.5  Ohm 


t 


100  Volte 

JLB 

_  1.25  Ohma 


0.5  Ohm 

AA/VWWWN 

FIG.  17 . 

power  used  will  therefore  be  the  same  in  either  case.  Con- 
sider the  circuit  shown  in  Fig.  17,  in  which  the  generator  G 
is  charging  the  battery  B.  Suppose  the  generator  develops 
an  e.m.f.  of  124  volts  and  the  battery  an  e.m.f.  of  100  volts; 
and  suppose  the  resistance  of  the  generator  is  0.75  ohm,  of 
the  battery,  1.25  ohms,  and  of  the  wires  leading  from  the 
generator  to  the  battery,  1.0  ohm,  making  a  total  resistance 
of  3.0  ohms.  The  resultant  e.m.f.  acting  around  the  circuit 
will  be  124-100=24  volts;  the  current  flowing  will  there- 
fore be  24/3=8  amperes.  Since  100  volts  is  used  to  send 
the  current  through  the  battery  against  its  back  e.m.f., 
this  back  e.m.f.  may  be  represented  by  a  resistance  of 
100/8  =  12.5  ohms  and  the  power  developed  will  be  82Xl2.5 


38  ELECTRIC  AND  MAGNETIC  CIRCUITS 

=  800  watts;  evidently,  the  same  result  would  be  obtained 
by  multiplying  the  battery  e.m.f.  by  the  current,  that  is 
8x100=800  watts. 

The  total  power  developed  in  this  circuit  is  8  X 124  =992 
watts,  of  which  82Xl  =64  watts  are  used  in  the  lead  wires; 
82  xO.75  =48  watts  are  used  in  the  resistance  of  the  generator, 
and  82Xl.25=80  watts  are  used  in  the  internal  resistance 
of  the  battery.  Since  the  current  is  flowing  against  the 
e.m.f.  of  the  battery  as  well  as  against  its  resistance  the 
potential  difference  at  the  terminals  of  the  battery  will  be 
100 +  (8xl. 25)  =110  volts;  the  potential  difference  at 
the  terminals  of  the  generator  is  124 -(8x0.75)  =118  volts, 
and  the  drop  in  the  two  lead  wires  is  8  volts. 

The  work  done  or  energy  developed  in  any  process  during 
a  given  time  is  equal  to  the  power,  or  rate  of  doing  work, 
multiplied  by  the  given  time;  that  is,  using  the  symbol  W 
for  work, 

W  =  Pt  =  EIt  (8) 

in  an  electrical  circuit.  But  7  is  the  rate  of  flow  of  elec- 
tricity and  therefore  It  is  the  quantity  of  electricity,  Q, 
which  flows  through  the  circuit  in  the  time  t.  We  may 
therefore  write 

W  =  EQ,  (9) 

which  tells  us  that  the  work  done  in  an  electrical  circuit  is 
equal  to  the  product  of  the  electromotive  force  and  the 
quantity  of  electricity  Q,  which  is  sent  through  the  circuit 
either  by  or  against  the  e.m.f.,  E.  If  the  e.m.f.  sends  the 
quantity  through  the  circuit,  work  is  done  on  the  circuit; 
if  the  quantity  is  sent  through  against  the  e.m.f.,  work  is 
done  by  the  circuit.  Equation  (9)  may  be  written  in  the 
form 

W 
E  =  -Q,  (10) 

which,  interpreted,  means  that  electromotive  force  is  equal 
to  the  work  done  in  moving  unit  quantity  of  electricity. 
That  is,  when  the  potential  difference  between  two  points 


ELECTRIC  CIRCUITS  39 

is  E  volts,  the  work  done  (in  joules)  in  moving  one  coulomb 
of  electricity  from  one  of  the  points  to  the  other  is  numer- 
ically equal  to  the  voltage  E. 

30.  The  Circular  Mil. — Most  of  the  conductors  used  in 
electrical  work  are  round  and  less  than  1  inch  in  diameter. 
In  order  to  avoid  the  use  of  decimals  in  expressing  the 
diameter  of  such  wires,  a  unit  known  as  the  mil  has  come 
into  very  general  use.     It  is  equal  to  1/1000  of  an  inch. 
Instead  of  saying  that  a  wire  is  0.4  of  an  inch  in  diameter, 
it  is  said  to  be  400  mils  in  diameter. 

The  cross-sectional  area,  in  square  units,  of  a  round  wire 
is,  of  course,  equal  to  the  square  of  its  diameter  multiplied 
by  7T/4;  thus  the  area  of  a  wire  400  mils  in  diameter  is 
160,000  times  ?r/4  or  125,700  square  mils.  In  order  to 
avoid  the  use  of  the  factor  7r/4,  another  new  unit,  known  as 
the  circular  mil,  has  come  into  general  use.  Its  value  is 
simply  the  area  of  a  circle  whose  diameter  is  1  mil;  it  is 
therefore  equal  to  0.7854  square  mil,  and  the  area  of  any 
circle  expressed  in  circular  mils  is  equal  simply  to  the  square 
of  its  diameter.  That  is,  if  the  diameter  of  a  circle  is  cl  mils, 
its  area  is  equal  to  the  sum  of  the  areas  of  d2  circles,  each 
of  whose  diameter  is  1  mil,  or,  d2  circular  mils;  the  area  of  a 
wire  400  mils  in  diameter  is  160,000  circular  mils.  See 
Standard  Handbook  for  Electrical  Engineers,  Section  4, 
Paragraphs  10  to  30  for  information  on  wire  gauges  and 
tables. 

31.  Specific  Resistance. — The  resistance  of  a  given  con- 
ductor depends  upon  the  material  of  which  it  is  composed 
and  upon  its  dimensions.     It  varies  directly  as  the  length 
and  inversely  as  the  cross-sectional  area.  Expressed  in  the 
form  of  an  equation,  the  resistance,  R,  of  wire  of  length, 
I,  and  area,  a,  is 

B-'i  (ID 

where  p  is  a  constant  expressing  the  value  of  the  resistance 
of  a  piece  of  the  given  material  1  unit  long  and  1  unit  in 


40  ELECTRIC  AND  MAGNETIC  CIRCUITS 

area.  The  constant  p  is  called  the  specific  resistance,  or 
resistivity  of  the  material.  A  wire  1  ft.  long  and  1  mil  in 
diameter  (i.e.,  1  circular  mil  in  area)  is  called  a  circular-mil- 
foot,  or  more  commonly,  a  "  mil-foot."  The  value  of  p 
varies  somewhat  with  temperature,  as  will  be  discussed  in 
the  next  paragraph.  Its  value  for  copper  is  10.37  ohms 
per  mil-foot  at  20°  C.  This  is  known  as  the  "  International 
Annealed  Copper  Standard  "  and  is  for  commercially  pure 
electrolytic  copper.  It  represents  the  most  recent  experi- 
mental determinations.  The  conductivity  of  copper  varies 
with  its  purity  and  its  physical  condition.  Copper  having 
the  above  value  of  specific  resistance  is  defined  by  inter- 
national agreement  as  having  100  per  cent  conductivity. 
Copper  of  98  per  cent  conductivity  would  have  a  specific 
resistance  of  10.37/0.98  =  10.58  ohms.  The  International 
Standard  differs  slightly  from  the  older  standard  (known 
as  Matthiessen's  Standard),  which  was  10.35  ohms.  The 
International  Standard  specific  resistance  for  copper  at  0°  C. 
is  9.556  ohms  per  mil-foot.  Values  of  specific  resistance 
for  other  metals  may  be  found  in  any  Electrical  Engineer's 
Handbook.  For  a  complete  table  of  the  properties  of  cop- 
per wire,  see  Standard  Handbook  Section  4,  Paragraph  50. 

32.  Effect  of  Temperature  on  Resistance. — The  resistance 
of  a  conductor  is  found  to  depend  upon  its  temperature  as 
well  as  upon  the  material  it  is  made  of.  The  pure  metals 
increase  in  resistance  as  the  temperature  increases.  The 
resistance  of  certain  alloys  increases  but  very  slightly  with 
temperature,  and  in  a  few  cases  even  decreases  slightly. 
The  resistance  of  salt  and  acid  solutions  and  of  carbon 
decreases  with  temperature. 

The  increase  of  resistance  of  a  given  wire,  due  to  increase 
in  temperature  is  proportional  to  the  initial  resistance  of  the 
wire  and  very  nearly  proportional  to  the  rise  in  temperature. 
That  is,  if  RQ  be  the  resistance  of  the  wire  at  some  standard 
temperature,  such  as  zero  Centigrade,  the  increase  in  resist- 
ance is  equal,  very  closely,  to  aRot,  where  t  is  the  rise  of 
temperature  above  zero,  and  a  is  a  constant  known  as  the 


ELECTRIC  CIRCUITS  •        41 

temperature  coefficient  of  the  given  material.  The  tempera- 
ture coefficient  may  be  defined  as  the  increase  of  resistance 
per  degree  Centigrade  per  ohm  of  resistance  at  the  initial 
standard  temperature.  The  resistance  of  the  wire  at  tem- 
perature t  is  therefore 

R  =  R0  +  R0at  =  Ro(l+at)  (12) 

The  constant  a,  is  not  the  same  for  different  initial  tem- 
peratures. The  International  Standard  Temperature  Coeffi- 
cient for  copper  of  100  cent  conductivity  is  as  determined 
by  experiment  0.00393  for  an  initial  temperature  of  20°  C., 
and  varies  directly  as  the  conductivity.  For  any  initial 
temperature,  t,  and  for  100  per  cent  conductivity  the  coeffi- 
cient is  a*  =  1  7234.5  +Z,  very  nearly.  It  is  convenient  for 
many  purposes  to  use  the  temperature  coefficient  corre- 
sponding to  0°  C.  as  the  initial  temperature.  The  value  of 
ao  for  copper  at  0°  C.  is  0.00427. 

The  formula  for  finding  the  resistance  at  some  tempera- 
ture, t',  when  the  resistance  at  some  other  temperature,  t, 
is  known,  is 

Rt.  =  Rt[l+at(l'-f)].  (13) 

For  copper,  this  reduces,  when  (1/234.5+0  is  substituted 
for  at,  to  the  form 


It  is  usual  in  electrical  testing  of  machinery'  to  determine 
the  temperature,  t',  of  windings  from  the  measurement  of 
their  resistances  at  a  known  temperature,  t,  and  at  the 
unknown  temperature,  t'.  For  this  calculation,  equation 
(14)  reduces  to  the  form, 

•p 

£'  =  ^(234.5+0-234.5.  (15) 

tit 

When  a  wire  is  carrying  current,  the  heat  generated  in  it 
due  to  its  resistance  raises  its  temperature,  and  the  temper- 
ature will  rise  until  the  rate  at  which  the  heat  is  radiated 


42  ELECTRIC  AND  MAGNETIC  CIRCUITS 

and  conducted  away  from  the  wire  is  equal  to  the  rate  at 
which  heat  is  generated  in  it.  The  rate  at  which  the  heat 
is  carried  away  depends  upon  the  surroundings  of  the  wire; 
that  is,  whether  it  is  bare  or  insulated,  and  whether  it  is 
strung  in  open  air  or  wound  up  in  a  coil,  and  if  in  a  coil, 
whether  the  coil  is  enclosed  or  exposed  or  immersed  in  oil. 
See  Standard  Handbook  for  Electrical  Engineers,  Section  3, 
Paragraph  22,  for  table  of  carrying  capacity  of  insulated 
wires  as  allowed  by  the  National  Board  of  Fire  Under- 
writers. When  the  wire  is  wound  in  a  coil,  the  carrying 
capacity  may  be  estimated  by  the  use  of  an  experimentally 
determined  constant  which  applies,  as  nearly  as  can  be 
judged,  to  the  conditions  as  to  exposure,  depth  of  winding, 
kind  of  insulation,  etc.  This  constant  is  expressed  as  the 
number  of  "  circular  mils  per  ampere,"  and  is  the  area  in 
circular  mils  which  the  wire  should  have  for  each  ampere  it 
is  to  carry,  in  order  that  the  temperature  shall  not  exceed  a 
safe  value.  For  a  temperature  rise  of  50°  C.  above  20°  C., 
the  required  circular  mils  per  ampere  will  vary  from  as  low 
as  600  for  shallow,  well  ventilated  coils,  to  as  high  as  2500 
for  deep  coils  and  little  ventilation. 

33.  KirchhofPs  Laws. — Two  laws  of  very  great  impor- 
tance in  connection  with  electric  circuits  were  first  clearly 
pointed  out  by  Kirchhoff,  and  have  been  proven  to  be  of 
universal  application. 

First  Law. — The  algebraic  sum  of  all  the  e.m.f's  acting 
in  a  chosen  direction  around  any  closed  circuit  is  equal  to 
the  algebraic  sum  of  all  the  resistance  drops  in  the  same 
direction  around  that  circuit. 

Second  Law. — The  sum  of  all  the  currents  which  flow 
up  to  any  point  in  a  circuit  is  equal  to  the  sum  of  all  the  cur- 
rents which  flow  away  from  that  point. 

The  significance  and  application  of  these  laws  under 
different  circuit  conditions  are  shown  in  the  following 
paragraphs. 

34.  Resistances  in  Series. — When  a  number  of  resistances 
are  connected  in  series,  the  same  current  flows  through  all  of 


ELECTRIC  CIRCUITS  43 

them,  and  the  total  resistance  is  equal  to  the  sum  of  the  indi- 
vidual resistances.  If  a  certain  current  is  flowing  through  a 
series  of  resistances,  the  net  e.m.f.  acting  in  the  circuit,  by 
KirchhofFs  first  law,  is  equal  to  the  sum  of  the  resistance 
drops,  or  to  the  product  of  the  sum  of  the  resistances,  and 
the  current.  That  is, 

E  =  RJ+I?2l+RzI  =  I(Ri+R2'+R3).  (16) 

If  there  is  more  than  one  e.m.f.  acting  on  the  circuit,  as,  for 
example,  the  back  e.m.f.  of  a  battery  or  of  a  motor,  then  the 
net  e.m.f.,  or  the  algebraic  sum  of  the  various  e.m.f  's,  is 
equal  to  the  sum  of  the  various  R  I  drops. 

35.  Resistances  in  Parallel.  —  When  two  or  more  resist- 
ances are  connected  in  parallel  between  two  points,  the 
current  flowing  through  each  resistance  is  equal  to  the  p.d. 
between  the  two  points  divided  by  that  particular  resistance, 
provided  that  there  are  no  sources  of  e.m.f.  connected  in 
series  with  any  of  the  resistances,  and  the  total  current  is, 
by  Kirchhoff's  second  law,  the  sum  of  the  currents  flowing 
in  the  various  paths.  It  is  many  times  convenient  to 
determine  the  value  of  a  single  resistance  which  is  equiva- 
lent to  the  several  resistances  in  parallel.  This  equivalent 
resistance  will  evidently  be  equal  to  the  p.d.  divided  by 
the  total  current  which  flows  between  the  two  points.  Its 
value  in  terms  of  the  individual  resistances  may  be  easily 
derived  as  follows  :  Let  the  total  current  be  represented  by  7, 
the  p.d.  by  E  and  the  individual  currents  and  resistances 
by  I  it  /2,  1  3,  etc.,  Ri,  R2,  #3,  etc.,  respectively. 
Then  we  may  write 


Therefore 


*-!- 


Blfiifii 


44  ELECTRIC  AND  MAGNETIC  CIRCUITS 

where  R  is  the  equivalent  resistance  of  Ri,  R2,  and  R3,  when 
connected  in  parallel.  That  is,  the  equivalent  resistance  of 
a  number  of  resistances  in  parallel  is  equal  to  the  reciprocal 
of  the  sum  of  the  reciprocals  of  the  individual  resistances. 
If  the  resistances  are  equal,  the  equivalent  resistance  will 
be  equal  to  the  value  of  one  divided  by  the  number  in 
parallel. 

The  reciprocal  of  a  resistance  is  called  its  conductance 
and  in  dealing  with  parallel  circuits  it  is  very  common  to  use 
conductances  instead  of  resistances.  The  unit  of  conduc- 
tance is  the  mho  (ohm  spelled  backward  and  pronounced 
mo).  The  symbol  used  for  conductance  is  G  and  a  circuit 
having  say  5  ohms  resistance,  is  said  to  have  a  conductance 
of  1/5,  or  0.2  mho.  By  Ohm's  Law,  I  =  E/R;  if  we  use 
G  instead  of  R,  it  is  7  =  E  xG.  From  the  equation  above,  it  is 
clear  that  the  equivalent  conductance  of  a  number  of  con- 
ductances in  parallel  is  equal  to  the  sum  of  the  individual 
conductances,  and  the  equivalent  resistance  is  the  reciprocal 
of  the  equivalent  conductance;  that  is, 

7  =GiE  +G2E  +G,E  =  E(G,  +£2  +£3)  (19) 

or 


36.  Series-parallel  Circuits.  —  In  the  case  of  a  mixed 
circuit,  such  as  that  shown  in  Fig.  18,  it  is  necessary,  first  of 
all,  to  find  the  equivalent  resistance  of  the  parts  in  parallel, 
and  then  add  the  result  to  the  resistances  in  series.  For 
example,  let  R2=5  ohms;  #3  =  10  ohms;  Ri=4  ohms; 
r,  the  resistance  of  the  battery  =3  ohms;  and  E}  the  e.m.f. 
of  the  battery  =6.2  volts;  what  will  be  the  total  current? 
The  conductance  of  R2  is  0.2  mho;  of  R3  is  0.1;  and  of  the 
combination,  0.3  mho;  the  equivalent  resistance  of  R2  and 
Rz  is  therefore  3.33  ohms.  The  total  resistance  of  the  cir- 
cuit is  then  3.33+4+3  =  10.33  ohms,  and  7  =  6.2/10.33  =0.6 
ampere.  The  p.d.  between  A  and  B  will  be  0.6x3.33=2 
volts,  the  current  through  R2  is  2/5=0.4  ampere,  and 


ELECTRIC  CIRCUITS 


45 


through  R3  is  2/10=0.2  ampere.  The  p.d.  in  Ri  is  0.6 
X4  =2.4  volts,  and  in  r  is  0.6  X3  =  1.8  volts;  the  sum  of  the 
various  p.d's  is  2+2.4  +  1.8  =  6.2,  and  is  equal  to  the  e.m.f. 
of  the  battery. 

To  further  illustrate  the  principles  used  in  solving  a 
series-parallel  circuit,  the  following  example  is  given:  Let 
the  circuit  be  made  up  as  shown  in  Fig.  19;  required  the  p.d. 
at  the  terminals  xy  of  RQ  when  the  p.d.  at  the  terminals 
A  B  of  the  battery  is  100  volts.  The  resistance  of  KxyN 
is  25+3+3=31  ohms,  which  is  in  parallel  with  20  ohms. 


A/VAA/W — i 
A  B 


FIG.  18. 


The  conductance  of  KxyN  is  1/31=0.03226;  of  R6  it  is 
1/20=0.05;  the  conductance  of  the  parallel  paths  between 
K  and  N  is  therefore  0.03226+0.05=0.08226  and  the 
equivalent  resistance  of  this  path  is  1/0.08226  =  12.15  ohms; 
this  equivalent  resistance  is  in  series  with  R±  and  R5;  the 
resistance  of  the  path  from  D  to  F  by  way  of  RQ  and  R9  is 
therefore  12.15+2+2  =  16.15  ohms,  and  its  conductance 
is  1/16.15=0.06192.  The  conductance  of  R3  is  1/19  = 
0.05263  and  of  the  parallel  circuit  between  D  and  F,  it  is 
0.06192+0.05263=0.11455.  The  equivalent  resistance  of 
the  parallel  path  between  D  and  F  is  therefore  1/0.11455  = 
8.73  ohms;  this  is  in  series  with  Ri  and  R2  so  that  the 


46 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


equivalent  resistance  of  the  entire  circuit  is  8.73  +  1.5  + 
1.5  =  11.73  ohms  and  the  total  current  is  100/11.73=8.53 
amperes. 

This  total  current  flows  from  A  to  D  then  divides;  the 
two  parts  join  again  at  F  and  flow  from  F  to  B.  The  drop  of 
potential  in  R\  is  "therefore  1.5x8.53  =  12.79  volts  and  like- 
wise the  drop  in  _R2  is  12.79  volts;  the  p.d.  between  D  and  F  is 
therefore  100  -  (2  X  12.79)  =  74.42.  The  current  which  flows 
from  D  to  F  by  way  of  RG  and  R»  is  therefore  74.42/16.15 
=  4.61  amperes.  The  current  in  R3  is  74.42/19=3.92. 
It  may  be  noted  that  3.92+4.61  gives  8.53,  which  is  the 


FIG.  19. 

total  current  as  previously  found,  and  thus  checks  the 
arithmetical  work.  The  drop  in  R4  and  also  in  R5  is 
4.61X2=9.22  volts,  so  that  the  p.d.  between  K  and  N  is 
74.42 -(2x9.22)  =55.98  volts.  The  current  in  KxyN 
is  55.98/31  =  1.81  and  in  RQ  it  is  55.98/20=2.8.  The  sum 
of  these  is  4.61  which  again  checks  with  the  value  found 
above  for  the  circuit  from  D  to  F  by  way  of  RQ  and  Rg. 
The  drop  in  R7  and  R8  is  2x3x1.81  =10.86  volts,  so  that 
the  p.d.  between  x  and  y  is  55.98-10.86=45.12  volts. 

If  the  above  problem  had  been  stated  by  giving  the  p.d. 
between  x  and  y  and  requiring  the  p.d.  between  A  and  B, 
it  would  not  have  been  necessary  to  calculate  the  equivalent 
resistance  of  the  circuit.  The  current  in  Rg  would  be  found 


ELECTRIC  CIRCUITS  47 

at  once,  then  the  drop  in  Ri  and  R&,  then  the  p.d.  between 
K  and  N  by  adding  these  drops  to  the  p.d.  between  x  and  y. 
Then  the  current  in  RQ  would  be  found  and  added  to  that  in 
RQ  giving  the  current  in  R±  and  R5',  then  the  drops  in  #4 
and  in  R5  would  be  found  and  added  to  the  p.d.  between 
K  and  N,  giving  the  p.d.  between  D  and  F.  Then  the 
current  in  Rs  would  be  found  and  added  to  that  in  R^ 
giving  the  total  current,  then  the  drop  in  Ri  and  R2  would 
be  found  and  added  to  the  p.d.  between  D  and  F,  giving 
the  p.d.  between  A  and  B.  This  general  problem  may  be 
met  in  practice  in  many  forms;  for  example,  instead  of  the 
resistances  R%,  RG,  and  RQ  being  given,  the  currents  may  be 
given;  or,  the  power  in  one  or  more  may  be  given;  or,  the 
p.d's  between  A  and  B  and  between  x  and  y  may  be  given 
and  the  resistances,  Ri,  R2,  R±,  Rd,  Ri,  and  R%  required. 
The  relations  between  power,  p.d.,  current  and  resistance 
must  all  be  kept  continually  in  mind. 

37.  Complex  Circuits. — There  is  a  certain  class  of  cir- 
cuits which  requires  the  application  of  Kirchhoff's  and  Ohm's 
Laws  in  a  somewhat  different  manner  from  that  used  in  the 
preceding  problems.  An  example  of  this  is  shown  in  Fig.  20. 
The  resistance  of  this  combination  cannot  be  found  by  the 
methods  given  above;  but  by  applying  KirchhofTs  two  laws, 
equations  can  be  written  by  which  the  unknowns  can  be 
found. 

Let  it  be  assumed  that  six  resistances,  n,  r^  r3,  r±,  r5, 
and  TQ  are  known  and  also  the  p.d.,  E,  between  the  ter- 
minals of  the  battery;  the  unknowns  are  the  six  currents 
in  1,  2,  3,  4,  5,  and  6.  It  will  be  noted  that  there  are  four 
points  in  the  circuit  shown  in  Fig.  20,  where  the  current 
divides.  Writing  the  equations  for  the  current  at  each  of 
these  points,  we  have: 

76  =  /i+/2  (point  a)  (21) 

/1=/,  +  74  (point  6)  (22) 

72  +  73=/5  (point  c)  (23) 

/6  (point  d)  (24) 


48 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


It  should  be  noted  that  the  current  in  path  r3  is  assumed  to 
flow  from  b  toward  c;  it  is  necessary  to  assume  a  direction 
for  the  current  in  all  paths  in  order  to  write  the  equations; 
if,  in  the  final  solution,  any  current  comes  out  with  a  nega- 
tive sign,  it  means  that  the  current  in  that  path  flows  in 
the  direction  opposite  to  that  assumed.  It  should  be  noted 
also  that  any  one  of  the  four  equations  written  above  may 
be  derived  mathematically  from  the  other  three  and  that 
therefore  only  three  of  the  equations  are  independent  of 


each  other  and  can  be  used  for  the  solution  of  the  unknown. 
To  write  the  equations  for  the  first  law,  note  that  there  are 
also  four  paths  for  which  equations  may  be  written,  namely, 
abdfa,  acdfa,  abca,  and  bcdb,  but  as  in  the  case  of  the  current 
equations,  only  three  of  them  are  independent.  There  are 
other  paths  which  can  be  traced  through,  but  they  also  are 
related  to  the  rest  so  that  if  used  they  would  reduce  to 
identities.  The  equations  for  the  four  paths  mentioned 
are, 

n/i  +7*4/4  +r6/6  =  E  (abdfa)  (25) 


ELECTRIC  CIRCUITS 


49 


=  E(acdfa)  (26) 

ri/i  +7*3/3  -7*2/2  =0(dbca)  (27) 

7*3/3  +7*5/5  —  7*4/4  =0(lcdb)  (28) 

We  thus  have  six  equations  from  which  to  find  six  unknowns. 
Two  of  the  equations,  one  from  each  set,  are  not  to  be  used, 
and  the  solution  of  the  problem  is  one  of  pure  mathematics. 
The  negative  sign  is  used  for  r2/2  in  equation  (27)  and  for 


7*4/4  in  equation  (28)  because  these  drops  are  opposite  in 
direction  from  the  other  two  in  their  respective  paths.  In 
some  cases,  other  e.m.f  s  may  be  placed  in  the  network,  as, 
for  instance,  in  with  r$  as  shown  in  Fig.  21.  In  such  case 
the  e.m.f.  equation  for  the  path  acdfa  would  be 

7-2/2  +7*5/5  +r6/6  =  E±e  (29) 

the  sign  in  front  of  e  depending  on  whether  the  e.m.f.  e  is 
with  or  against  the  e.m.f.  E.  For  the  circuit  shown,  it 
would  have  a  negative  sign. 


50 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


The  equation  for  path  bcdb  would  be 

r3/3 +7*5/5 -r4/4  =  -e.  (30) 

When  two  e.m.f  s  are  connected  in  parallel  as  in  Fig.  22, 
the    current  will  divide  equally  between   the  two  sources 


v/W\AAAV\AVVVWUV 


FIG.  22. 


when  the  e.m.f  s  are  equal  and  the  resistances  in  the  two 
paths  are  equal.  If  either  or  both  of  these  are  unequal,  the 
current  may  not  divide  equally  and  may  flow  in  either  direc- 
tion through  one  of  the  sources,  depending  on  the  relative 


-MAW- 


J". 

FIG.  23. 

values  of  EI,  EZ,  and  the  resistances  in  the  three  branches. 
The  solution  of  such  a  problem  involves  .the  writing  of  the 
equations  for  Kirchhoff  s  Laws  and  solving  them  for  the 
unknowns. 

Another  example  of  a  complex  circuit  is  the  circuit  fre- 
quently used  in  distributing  electrical  power,  and  known  as 


ELECTRIC  CIRCUITS 


51 


the  three-wire  circuit.  It  is  represented  in  simple  form  in 
Fig.  23.  The  middle  wire  (r3)  is  frequently  called  the  neutral 
wire  and  the  current  in  it  may  flow  in  either  direction,  or, 
if  the  loads  R±  and  R5  are  perfectly  balanced,  no  current 
will  flow  in  it. 

38.  The  Wheatstone  Bridge. — In  Fig.  24  is  shown  a  net- 
work similar  to  that  of  Fig.  20,  but  R%  is  replaced  by  a 
galvanometer  and  resistance  ?v  If  the  resistances  n,  r%,  r3 


and  rx  are  given  such  values  that  no  current  flows  through  rg 
and  the  galvanometer,  then  the  points  b  and  c  must  be  at 
the  same  potential,  and  the  p.d.  from  a  to  b  must  be  the 
same  as  that  from  a  to  c;  likewise,  the  p.d.  from  b  to  d 
must  be  the  same  as  that  from  c  to  d.  Also  the  current  /i 
is  the  same  as  7  2  and  73  is  the  same  as  IX)  since  no  current 
flows  in  TV  Therefore  we  may  write 


and 


r2Ii=rxh. 


(31) 
(32) 


52  ELECTRIC  AND  MAGNETIC  CIRCUITS 

From  these  equations,  we  get 

s-s- 

This  relation  is  used  in  the  measurement  of  resistance 
and  the  arrangement  is  called  a  Wheatstone  Bridge.  There 
are  various  forms  of  Wheatstone  Bridges,  among  them  being 
the  Box  Bridge  and  the  Slide-wire  Bridge.  In  the  Box  Bridge 
there  are  three  sets  of  adjustable  resistances  within  the  box, 
which  may  be  represented  by  n,  r2  and  r3  in  Fig.  24.  Bind- 
ing posts  are  provided  so  that  a  battery  may  be  connected  to 
points  a  and  d,  a  galvanometer  between  points  b  and  c,  and 
an  unknown  resistance  between  c  and  d.  Sets  n  and  r2 
are  usually  called  the  ratio  arms  of  the  bridge  and  generally 
each  one  has  a  10-ohm,  a  100-ohm,  and  a  1000-ohm  coil, 
any  one  of  which  may  be  connected  in,  so  that  the  possible 
ratios  of  r\  to  r%  are  0.01,  0.1,  1,  10,  and  100.  Sometimes  a 
1-ohm  coil  is  also  put  in  each  set,  thus  giving  possible  ratios  of 
0.001  and  1000.  Set  r3  consists  of  a  number  of  coils,  of 
values  ranging  from  1  to  500  or  1  to  5000,  and  so  arranged 
that  steps  of  1  ohm  can  be  made  from  1  to  1111  or  from  1  to 
11111. 

An  unknown  resistance  being  connected  between  c  and  dy 
the  ratio  arms  and  the  resistance  r3  are  adjusted  until  the 
galvanometer  shows  no  deflection  when  its  circuit  is  repeat- 
edly opened  and  closed.  Then  the  unknown  resistance  is 
calculated  from  the  relation 


(34) 


In  the  slide-wire  bridge  (see  Fig.  25)  a  piece  of  bare  wire 
of  uniform  size  is  stretched  between  two  points  over  a  scale, 
usually  one  meter  long.  A  known  resistance  being  con- 
nected between  a  and  6,  and  an  unknown  resistance  between 
b  and  d,  the  sliding  contact  c,  is  moved  along  until  there  is 
no  deflection  in  the  galvanometer.  When  this  condition  is 


ELECTRIC  CIRCUITS 


53 


obtained  the  resistances  are  related  to  each  other  in 
proportion  ab  :  bd  ::  ac  :  cd]  whence 

rd 


the 


(35) 


But  the  resistances  of  ac  and  cd  have  the  same  ratio  as  the 
corresponding  portions  of  the  slide  wire,  which  is  therefore 
easily  determined  from  the  readings  on  the  scale.  Some- 
times a  telephone  receiver  is  used  in  place  of  the  galvanometer 
for  determining  the  condition  of  balance,  and  answers  very 
'well  for  work  not  requiring  great  accuracy. 


H 


FIG.  25. 

39.  The  Potentiometer. — A  method  of  connection  fre- 
quently useful  in  electrical  testing  work  is  known  as  the 
potentiometer  connection.  It  consists  of  a  resistance  which 
is  accessible  at  all  points  or  at  frequent  intervals  along  its 
length,  connected  to  a  source  of  e.m.f.  which  is  higher  than 
the  e.m.f.  which  is  desired  or  which  is  to  be  measured. 
Fig.  26  illustrates  the  method  of  connection  for  obtaining 
any  desired  p.d.  between  the  points  a  and  c  within  the  limits 
of  the  e.m.f.  impressed  on  ab.  A  sliding  contact  is  used  at 
c  and,  as  it  is  moved  from  a  toward  b,  the  p.d.  increases  from 
zero  to  the  full  value  between  a  and  b.  The  wire  ab  is 
called  the  potentiometer  wire.  Suppose  the  resistance  of  ab 
is  200  ohms  and  of  R  is  20  ohms;  required  a  p.d.  of  24  volts 
between  a  and  c  when  the  p.d.  between  a  and  b  is  100  volts. 


54 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


The  current  in  R  will  be  24/20  =  1.2  amperes;  the  p.d. 
between  c  and  b  will  be  100—24  =  76  volts.  The  current 
/cft  =  /ac  +  1.2;  also  rac/ac  =  24;  rc&/c6  =  76;  and  rac+rc6=200. 
The  solution  of  these  equations  will  give  the  values  of  the 
four  unknowns,  7ac,  7C&,  rac,  and  rcb. 

If  in  the  circuit  aRc,  a  battery  be  placed  so  that  its 
e.m.f.  opposes  the  flow  of  current  through  that  circuit,  there 
will  be  some  point  c  where  the  e.m.f.  of  this  second  battery 
will  just  balance  the  p.d.  in  the  potentiometer  wire  between 


^AAAA^AAAAAAAAA/^AAAAAAA^VV\VvV  b 


FIG.  26. 


a  and  c  and  no  current  will  flow  in  the  circuit  aRc.  This 
principle  is  used  for  comparing  the  values  of  two  e.m.f  s. 
A  known  e.m.f.  (a  standard  cell),  is  connected  between 
a  and  c,  Fig.  27,  and  the  point  c  is  found  where  no  current 
flows  through  the  galvanometer  G\]  the  e.m.f.  to  be  meas- 
ured is  connected  between  a  and  d  and  the  point  d  is  found 
such  that  no  current  flows  through  the  galvanometer  6^2. 
In  practice,  switching  arrangements  are  provided  by  means 
of  which  e  is  connected  in  the  place  of  es,  the  same  gal- 
vanometer is  used,  and  a  slider  is  used  to  find  the  positions 
c  and  d.  The  current  has  the  same  value  throughout  the 


ELECTRIC  CIRCUITS 


55 


wire  and  the  p.d.  between  a  and  c  is  to  the  p.d.  between 
a  and  d  as  the  resistance  of  the  potentiometer  wire  between 
a  and  c  is  to  the  resistance  between  a  and  d.  These  resist- 
ances may  be  known  and  their  ratio  is  the  ratio  of  es  to  e] 
or,  the  potentiometer  wire  may  be  of  uniform  cross-section 
so  that  the  resistance  of  any  portion  is  proportional  to  the 


^/wsA/wwww6 

d 


FIG.  27. 

length  of  that  portion;  in  this  case  the  ratio  of  the  lengths 
ac  and  ad  is  the  required  ratio  of  es  to  e. 

40.  Ammeters  and  Voltmeters  for  Direct  Currents.— 
The  D' Arson val  type  of  meter  was  briefly  described  in  the 
last  paragraph  of  Article  22.  Fig.  28  shows  a  plan  view  of 
a  commercial  meter  of  this  type.  The  coil,  CC,  is  rectangular 
in  shape  as  shown  in  Fig.  29.  One  end  of  the  coil  is  con- 
nected to  the  inner  end  of  the  upper  spiral  spring,  s';  and 
the  other  end  of  the  coil  is  connected  to  the  inner  end  of  the 
lower  spiral  spring,  s".  The  outer  ends  of  these  spiral 


56 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


springs  serve  as  terminals  for  leading  the  current  to  and  from 
the  coil.  The  coil  is  pivoted  concentrically  with  the  pole 
pieces,  PP'}  Fig.  28,  which  are  attached  to  the  poles  of  the 
permanent  magnet  MM.  A  cylindrical  soft  iron  core^  /, 
occupies  the  space  inside  the  coil,  so  that  the  air  gaps,  gg,  in 
which  the  sides  of  the  coil  move,  are  uniform  in  length,  thus 
producing  a  uniform  magnetic  flux  density. 

A  typical   value   of  the   current   required  to   produce 
enough  torque  to  swing  the  needle  over  the  full  scale  (80° 


FIG.  28. 


i  i  i  i  f'i~rn 


I 

FIG.  29. 


to  90°)  is  0.01  ampere;  the  corresponding  typical  value  of 
the  voltage  required  to  send  this  current  through  the 
resistance  of  the  coil  and  springs  is  0.05  volt  (or,  50  milli- 
volts). The  resistance  of  the  coil  and  springs  in  this  case 
would  be  5  ohms. 

To  adapt  this  arrangement  to  the  measurement  of  large 
currents  or  voltages,  shunts  or  multipliers  must  be  used. 
To  illustrate,  suppose  a  meter  is  desired  to  measure  10 
amperes  at  full  scale  deflection.  This  is  1000  tunes  the 
current  allowable  through  the  coil;  therefore  a  resistance 
must  be  connected  in  parallel  with  the  coil,  which  will  carry 


ELECTRIC  CIRCUITS 


57 


9.99  amperes  when  the  coil  is  carrying  0.01  ampere.  The 
resistance  of  the  shunt  Rs  would  be,  in  this  case  1/999 
of  5  ohms.  The  connection  is  shown  in  Fig.  30.  The  scale 
of  the  meter  would  be  marked  to  show  the  total  current 
flowing,  that  is,  10  amperes  at  full  deflection. 

To  use  the  same  coil  for  measuring,  say  150  volts  at  full 
deflection,  it  would  be  necessary  to  put  in  series  with  the  coil 


FIG.  30. 


FIG.  31. 


a  resistance  of  such  value  that  150  volts  would  produce  0.01 
ampere  through  the  meter  at  full  scale  deflection.  The 
total  resistance  would  be  150/0.01,  or  15,000  ohms;  the 
multiplier,  or  additional  resistance,  Rm  would  have  to 
be  14,995  ohms.  This  connection  is  shown  in  Fig.  31.  In 
this  case,  the  scale  of  the  meter  would  be  marked  in  volts, 
from  0  to  150. 


CHAPTER  IV 
ELECTROMAGNETISM 

41.  Flux-linkage  and  Electromotive  Force. — Whenever 
an  electric  circuit  and  a  magnetic  circuit  are  so  related 
that  the  magnetic  lines  of  force  pass  through  and  around 
the  electric  circuit  like  two  adjacent  links  of  a  chain, 
the  two  circuits  are  said  to  be  interlinked.  When  all  the 
lines  of  force  in  a  given  field  interlink  with  all  the  turns  of  a 
coil  of  wire,  the  total  number  of  linkages  is  equal  to  the 
product  of  the  number  of  turns  in  the  coil  and  the  number  of 
lines  of  force.  When  the  interlinkage  is  not  complete,  see 
Fig.  8,  the  total  number  of  linkages  is  equal  to  the  sum  of 
all  the  lines  that  link  each  turn;  that  is,  one  linkage  is  one 
line  linking  with  one  turn. 

It  was  discovered  by  Faraday  that  when  from  any 
cause  the  number  of  linkages  in  a  circuit  changes,  there  is 
induced  in  the  circuit  an  electromotive  force  which  is  pro- 
portional to  the  rate  at  which  the  linkages  change.  The 
rate  of  change  of  linkage  is  frequently  called  rate  of  cutting 
lines  of  force,  since,  whenever  the  number  of  linkages 
changes,  either  lines  of  force  must  cut  across  the  wire  con- 
stituting the  circuit,  or  the  wire  must  cut  across  the  lines 
of  force.  In  this  connection,  it  should  be  noted  that  lines 
of  force  may  be  cut  by  a  wire  without  changing  the  linkage, 
and  therefore,  without  producing  an  e.m.f .  at  the  terminals 
of  the  wire.  For  example,  if  a  coil  of  wire  is  moved  within 
a  uniform  magnetic  field  in  such  a  direction  that  its  plane 
does  not  change  in  direction  with  respect  to  the  field,  both 
sides  of  the  coil  will  cut  lines  of  force,  but  the  number  of 
lines  passing  through  the  coil  will  not  be  changed  so  long  as 
the  whole  coil  remains  in  a  field  of  uniform  strength.  The 

58 


ELECTRON  AGNETISM  59 

linkage  with  each  half  of  the  coil  changes,  or  in  other  words, 
each  half  of  the  coil  cuts  lines  of  force,  and  equal  e.m.f  s 
are  generated  in  the  two  halves  but  these  two  e.m.f  s  are 
found  to  be  oppositely  directed  around  the  coil,  so  that  they 
balance  each  other,  and  the  resultant  e.m.f.  generated  in  the 
coil  is  zero. 

When  an  e.m.f.  is  generated  or  induced  in  the  manner 
mentioned  above  and  the  circuit  is  closed  so  that  current 
flows,  electric  power  will  be  developed  in  the  circuit.  Since 
power  cannot  be  created  out  of  nothing,  the  question  at 
once  arises,  whence  comes  this  power  and  from  what  kind  of 
power  is  it  transformed?  The  answer  is,  that  when  current 
flows  in  a  wire,  and  that  wire  is  in  a  magnetic  field,  a  mechan- 
ical force  is  exerted  on  the  wire  tending  to  push  it  sideways 
through  the  field;  this  is  the  reacting  force  against  which 
the  wire  must  be  moved  by  mechanical  means  through  the 
field  in  order  to  produce  the  electric  power.  That  is, 
mechanical  power  is  expended  in  moving  the  wire  through 
the  field,  and  this  mechanical  power  is  transformed  into 
electrical  power  by  virtue  of  the  e.m.f.  generated  and  the 
current  which  flows.  This  is  the  principle  of  electric  gen- 
erators. Note  that  no  power  is  required  to  generate  an 
e.m.f.  in  a  wire  if  no  current  is  flowing  in  the  wire. 

Suppose,  now,  that  a  wire  be  placed  in  a  magnetic  field, 
and  current  be  sent  through  it  by  an  external  source  of  e.m.f. 
The  force  exerted  on  the  wire  by  the  field  will  cause  it  to 
move  and  develop  mechanical  power;  this  power  must  be 
supplied  by  the  electric  circuit,  and  in  order  to  do  this  there 
must  be  a  reacting  force  in  the  electric  circuit  against  which 
the  current  is  forced  to  flow.  When  the  wire  moves  through 
the  magnetic  field  an  e.m.f.  is  generated,  which  opposes  the 
flow  of  current;  this  e.m.f.  is  called  back  or  counter  e.m.f.  and 
is  the  reacting  force  against  which  the  work  is  done.  This 
is  the  fundamental  principle  of  the  action  of  an  electric 
motor.  Note  that  there  will  be  some  power  required  to 
supply  the  heat  losses  in  the  resistance  of  wires. 

It  is  of  extreme  importance  that  this  matter  of  the  react- 


60  ELECTRIC  AND  MAGNETIC  CIRCUITS 

ing  forces  which  are  exerted  when  electrical  energy  is  trans- 
formed to  mechanical  energy,  or  vice  versa,  be  thoroughly 
understood.  Therefore,  the  principles  discussed  in  the 
preceding  paragraphs  are  here  restated  in  somewhat  different 
form.  Th*at  there  must  be  a  reacting  force  follows  directly 
from  a  broad  interpretation  of  Newton's  third  law  of  motion, 
which  is,  that  there  can  be  no  action  without  an  equal  and 
opposite  reaction.  The  application  of  the  law  to  electric 
circuits  was  discovered  by  Lenz,  and  the  statement  that  an 
induced  current  always  opposes  the  action  which  produces 
it,  is  known  as  Lenz's  Law.  In  the  case  where  any  part  of 
a  closed  electric  circuit  is  moved  across  a  magnetic  field,  or  a 
magnetic  field  is  moved  across  any  part  of  a  closed  electric 
circuit,  it  has  been  discovered  that  an  e.m.f.  is  generated, 
an  electric  current  flows,  and  work  is  done.  To  do  this 
work  requires  the  application  of  a  mechanical  force  to  move 
the  wire  or  the  magnetic  field  and  the  reacting  force  is  dis- 
covered to  be  an  unseen  force  exerted  by  the  magnetic  field 
upon  the  wire.  In  the  case  where  an  electric  current  is  sent 
through  a  wire  which  lies  across  a  magnetic  field,  it  has  been 
discovered  that  the  wire  or  the  field  will  move  and  do 
mechanical  work.  To  do  this  work  requires  the  application 
of  an  e.m.f.  to  send  the  current  through  the  wire  and  the 
reacting  force  is  discovered  to  be  an  induced  e.m.f.  which 
opposes  the  flow  of  current. 

42.  Relation  of  Induced  e.m.c.  to  Rate  of  Change  of 
Linkage. — By  the  help  of  the  principles  just  discussed,  we 
may  derive  the  fundamental  relation  between  the  value  of 
an  induced  e.m.f.  and  the  rate  of  change  of  linkages.  Con- 
sider a  simple  case  like  that  shown  in  Fig.  32  which  repre- 
sents a  straight  wire  AB  moving  at  right  angles  across  a 
magnetic  field.  The  wire  D  KNG  is  supposed  to  be  station- 
ary and  its  plane  is  at  right  angles  to  a  uniform  magnetic 
field,  the  lines  of  force  being  perpendicular  to  the  paper  and 
represented  in  cross-section  by  the  dots.  The  wire  AB  is 
supposed  to  slide  toward  the  left  along  the  wires  DK  and 
GN  at  a  uniform  velocity  of  v  centimeters  per  second; 


ELECTRON  A  GNETISM 


61 


if  the  wires  D  K  and  GN  are  I  centimeters  from  each  other, 
the  change  in  linkage  will  be  Hlv  lines  per  second  where  H 
is  the  intensity  of  the  field,  in  lines  per  square  centimeter. 
The  movement  of  the  wire  AB  will  generate  an  e.m.f.  of 
e  volts,  a  current  of  i  amperes  will  flow  through  the  circuit 
and  work  will  be  done  at  the  rate  of  ei  joules  per  second,  or 
eiXlQ7  ergs  per  second.  A  mechanical  force  Fm,  will 
therefore  be  required  to  move  the  wire.  The  reacting  force 
Fr,  will  be  that  exerted  by  the  magnetic  field  on  the  wire, 
and  will  be  equal  to  Hli/W  dynes,  as  already  shown  in 
Article  19.  The  rate  at  which  work  is  done  against  this 
force  will  be  ,Hliv/\Q  ergs  per  second.  The  rate  at  which 


K 


FJux  downward  . 


m 


N 


D 


B 


FIG.  32. 


mechanical  work  is  done  on  the  wire  in  moving  it  must  be 
equal  to  the  rate  at  which  electrical  work  is  done  by  the 
e.m.f.  which  is  generated  as  a  result  of  moving  it.  There- 
fore, we  may  write  the  equation 

(36) 


eilO7 


or 


10 

Hlv 
108' 


(37) 


That  is,  the  e.m.f.  in  volts,  is  equal  to  the  rate  at  which  the 
linkages  change,  divided  by  108.  If  the  distance  moved 
through  in  time  t  be  called  s,  then  v=s/t,  and 

Hh       *  (38) 


62  ELECTRIC  AND  MAGNETIC  CIRCUITS 

where  <£  is  the  total  change  in  linkage  during  the  time  t,  and 
is  equal  to  His,  since  Is  is  the  area  swept  over,  and  H  is  the 
number  of  lines  per  square  centimeter.  The  expression 
(0/0  is  evidently  a  rate  of  cutting  lines  of  force;  if  this  rate 
is  not  constant,  due  either  to  non-uniform  field,  or  a  non- 
uniform  velocity,  the  e.m.f.  will  vary  from  instant  to 
instant,  and  the  expression  must  be  put  into  differential 
form,  as 


If  a  coil  of  wire  having  N  turns  be  moved  in  a  magnetic 
field  and  the  flux  which  passes  through  the  coil  changes  by 
an  amount  d(j>,  in  the  time  dt,  then  the  e.m.f.  will  be 

' 


If  the  linkage  is  not  complete,  but  the  actual  change  of  link- 
age can  be  represented  by  d$'  =kNd(j),  where  k  is  some 
factor  less  than  unity,  then 


This  is  the  fundamental  equation  for  an  induced  e.m.f.  and 
is  universal  in  its  application.  It  should  be  noted,  however, 
that  it  gives  the  instantaneous  value  and  that  the  average 
value  over  an  extended  tune  will  depend  upon  the  average 
rate  at  which  the  linkages  change  with  time,  see  equation 
(38).  The  application  of  these  equations  to  electrical 
machinery  will  be  taken  up  in  later  courses.  The  important 
things  to  learn  here  are  that  at  any  instant  an  induced  e.m.f. 
is  equal  to  the  time  rate  of  change  of  flux  linkage  with 
the  circuit  at  that  instant,  and  that  a  rate  of  change  of  108 
linkages  per  second  gives  1  volt  of  electromotive  force.  The 
e.m.f.  generated  by  a  rate  of  change  of  1  linkage  per  second 
is  called  an  abvolt;  1  volt  is  therefore  equal  to  108  abvolts. 
It  is  desirable  also  at  this  time  to  learn  a  rule  for  determining 
the  direction  of  an  induced  e.m.f.  The  so-called  "  Left- 


ELECTROMAGNETISM  63 

hand  Rule"  has  already  been  given  for  determining  the 
direction  of  the  force  action  upon  a  wire  in  a  magnetic  field. 
Since  the  direction  of  motion  required  to  produce  an  e.m.f. 
is  opposite  to  that  of  the  force  action  of  the  current  pro- 
duced by  such  e.m.f.,  it  follows  that  if  the  forefinger  of  the 
right  hand  point  in  the  direction  of  the  flux,  and  the  thumb 
at  right  angles  to  the  forefinger,  point  in  the  direction  of 
motion  of  the  wire  with  respect  to  the  flux,  the  middle  finger, 
at  right  angles  to  both,  will  point  in  the  direction  of  the 
induced  e.m.f.  This  is  known  as  the  "  Right-hand  Rule." 
It  must  be  noted  that  the  thumb  must  point  in  the  direction 
of  the  motion  of  the  wire  with  respect  to  the  flux;  that  is,  if 
the  flux  is  moving,  say  toward  the  left,  the  relative  motion 
of  the  wire  with  respect  to  the  flux  is  toward  the  right. 

43.  Work  Done  When  an  Electric  Wire  Cuts  a  Magnetic 
Field. — The  force  in  dynes  exerted  on  an  electric  wire  I  centi- 
meters long  and  carrying  a  current,  7  amperes,  when  the  axis 
of  the  wire  makes  an  angle  0  with  the  direction  of  a  magnetic 
field  of  strength  H,  has  been  shown  in  Article  19  to  be  equal 
to  I  HI  sin  0/10.     If  the  wire  is  moved  a  distance  s  against 
this  force  and  the  direction  of  motion  makes  an  angle  a  with 
the  direction  of  the  force,  the  work  done  will  be  I  His  sin  0 
cos  a/10  ergs;  but  His  sin  0  cos  a  is  equal  to  </>,  the  number 
of  lines  of  force  cut  by  the  wire;   therefore,  the  work  W, 
done  when  a  wire  cuts  a  field,  is  equal  to  07/10,  the  product 
of  the  flux  cut  and  the  current  in  the  wire,  divided  by  10 
when  7  is  in  amperes.     If  the  electric  circuit  consists  of  a 
coil  of  wire  of  N  turns,  and  the  total  amount  of  cutting  of 
flux,  or  change  in  linkages,  is  4>N,  then  the  work  done  will  be 

TF  =  0N7/10ergs.  (42) 

44.  Number  of  Lines  of  Force  Issuing  from  a  Unit 
Magnet  Pole. — A  unit  magnet  pole  and  unit  strength  of 
magnetic  field  have  already  been  defined.     It  follows  from 
these  definitions  that  there  is  a  field  intensity  of  one  line 
per  square  centimeter  at  1  cm.  distance  from  a  unit  magnet 
pole;    therefore,  since  there  are  4r  square  centimeters  of 


64  ELECTRIC  AND  MAGNETIC  CIRCUITS 

surface  in  the  sphere  of  unit  radius  surrounding  a  unit  pole, 
there  will  be  a  total  of  4?r  lines  of  force  issuing  from  a  unit 
pole. 

45.  The  Field  Intensity  around  a  Long  Straight  Wire.— 
Consider  a  closed  electric  circuit,  consisting  in  part  of  a  long 
straight  wire  and  with  the  rest  of  the  circuit  far  enough 
removed  from  the  straight  part  that  the  flux  due  to  the  rest 
of  the  circuit  will  be  negligible  hi  the  vicinity  of  the  straight 
part.  The  magnetic  lines  of  force  surrounding  the  straight 
part  of  the  wire  will  then  be  concentric  circles  around  the 
center  of  the  wire  and  with  their  planes  at  right  angles  to 
the  wire.  Suppose  a  unit  pole  to  be  carried  once  around 
the  wire  along  one  of  the  concentric  lines  of  force;  each  of 
the  4?r  lines  of  force  from  the  pole  will  cut  the  circuit  and  the 
work  done  will  be  0.47r7,  when  /  is  in  amperes,  according  to 
equation  (42).  The  distance  moved  through  by  the  unit 
pole  is  2-n-x,  where  x  is  the  radius  of  the  particular  path 
followed.  Therefore,  the  force,  that  is,  the  intensity  of  the 
field  along  this  path,  is 


<«> 


This  is  an  important  law;  it  shows  that  the  intensity  of 
magnetic  field  produced  at  any  given  distance  from  a  long 
straight  wire  by  a  current  in  it  is  directly  proportional  to 
the  value  of  the  current  and  inversely  proportional  to  the 
radial  distance  from  the  center  of  the  wire  to  the  given  point. 
46.  Force  Exerted  between  Two  Parallel  Wires.  —  When 
two  wires  are  parallel  and  current  flows  through  them,  there 
will  be  a  force  exerted  between  them;  for  each  wire  produces 
a  field  which  is  at  right  angles  with  the  other  wire  and  there- 
fore exerts  a  force  on  it.  Suppose  the  wires  are  d  centi- 
meters apart  and  the  currents  are  /'  and  I"  amperes 
respectively,  then  the  field  intensity  produced  by  the  first 
wire  at  the  second  wire  is  0.27'/d;  this  field,  acting  on  the 
second  wire  produces  a  force  of  0.021'!"  /d  dynes  per  cen- 


ELECTROMAGNETISM  65 

timeter  of  length  of  wire.  Likewise,  the  second  wire  pro- 
duces a  field  of  0.27" /d  at  the  first  wire  and  this  field  exerts 
a  force  of  0.021'!" /d  dynes  per  centimeter  of  wire.  Each 
of  these  forces  is  the  reacting  force  of  the  other,  and  the 
attraction  or  repulsion  between  the  wires  is  therefore 
0.027 'I"/d  dynes  per  centimeter.  An  application  of  the 
"  left-hand  rule  "  will  show  that  the  force  will  tend  to 
draw  the  wires  together  when  the  two  currents  are  in  the 
same  direction  and  will  tend  to  push  them  farther  apart 
when  the  currents  are  opposite  in  direction.  When  the  two 
currents  are  equal,  then  the  force  between  the  wires  is 
proportional  to  the  square  of  the  current.  An  application 
of  the  law  just  discussed  is  to  be  found  in  the  construction 
of  Electrodynamometer  instruments. 

47.  Field  Intensity  at  the  Center  of  a  Coil  of  Large 
Radius. — It  has  been  proven  experimentally  that  the 
intensity  of  the  magnetic  field  which  emanates  from  a  mag- 
net pole  varies  inversely  as  the  square  of  the  distance  from 
the  pole;  and  since  the  field  intensity  is  taken  as  unity  at  a 
distance  of  1  cm.  from  a  unit  pole,  it  follows  that  the  field 
intensity  at  a  distance  of  r  centimeters  from  a  pole  of  m 
units  strength  is  ra/r2.  Since  the  intensity  of  a  field  H,  is 
measured  by  the  force  it  exerts  on  unit  pole,  the  force 
exerted  on  a  pole  of  m  units  strength  by  a  field  of  intensity 
H,  will  be  equal  to  mH.  The  force  exerted  on  a  wire  at 
right  angles  to  a  uniform  field  has  been  shown  to  be  equal 
to  the  product  of  the  field  intensity  the  strength  of  the  cur- 
rent and  length  of  wire.  If  a  coil  of  wire  has  Z  turns  of 
radius  r,  the  total  length  of  wire  is  2-n-rZ,  and  if  a  current,  /, 
be  sent  through  it,  a  certain  field  intensity,  H  will  be  pro- 
duced at  the  center  of  the  coil.  If  now  a  magnet  pole  of 
strength  m  be  placed  at  the  center  of  the  coil,  the  force 
exerted  upon  it  by  the  field  produced  by  the  coil  must 
equal  the  force  exerted  on  the  wire  by  the  field  produced 
by  the  pole;  that  is,  if  H  is  the  field  intensity  produced  at 
the  center  of  the  coil  by  the  current  in  it  and  m/r2  is  the 
intensity  of  field  produced  at  the  wire  by  the  pole,  then 


66  ELECTRIC  AND  MAGNETIC  CIRCUITS 

mH  =Q.2TrrZIm/r2,  since  each  of  these  forces  is  the  reaction 
of  the  other.  Therefore, 

H^.  (44) 

It  must  be  observed  that  this  value  of  intensity  holds  only 
at  the  center  of  the  coil,  and  also  that  it  holds  only  for  a  coil 
whose  radius  is  quite  large  in  comparison  with  the  radial 
depth  and  the  breadth  of  the  bundle  of  wires  making  up 
the  coil.  This  is  because  the  reasoning  is  based  on  the 
assumption  that  all  of  the  wires  may  be  considered  as  being 
equally  distant  from  the  pole  which  is  located  at  the  center 
of  the  coil. 

48.  Magnetomotive  Force. — The  ability  of  an  electric 
circuit  to  produce  magnetic  flux  is  called  its  magnetomotive 
force,  or  m.m.f.,  just  as  the  ability  of  a  battery  to  produce 
an  electric  current  is  called  its  electromotive  force,  or  e.m.f. 
The  measure  of  a  m.m.f.  is  taken  as  the  work  which  would 
have  to  be  done  hi  moving  a  unit  magnet  pole  from  any 
point  through  any  path  which  links  the  electric  circuit  back 
to  the  same  point  against  the  magnetic  force  produced  by 
the  current.  This  is  analogous  to  the  measure  of  an  e.m.f. 
as  expressed  in  equation  (10).  Suppose  the  electric  circuit 
to  consist  of  a  coil  of  N  turns  of  wire  and  to  carry  a  current 
of  I  amperes;  when  a  unit  pole  is  moved  from  any  point, 
through  the  coil,  and  back  to  the  starting  point,  each  of 
the  4?r  lines  of  force  issuing  from  this  unit  pole  will  cut  each 
of  the  N  turns  of  the  coil.  Therefore  the  total  cutting  of 
flux  will  be  47TJV  and  the  work  'done  will  be  OAwNI  ergs. 
The  product,  QAirNI,  is  the  general  expression  for  the 
magnetomotive  force  of  an  electric  circuit,  and  is  of  the 
greatest  importance  in  electrical  engineering.  The  funda- 
mental unit  of  m.m.f.  is  called  a  gilbert;  it  is  the  m.m.f.  which 
will  produce  a  field  intensity  of  one  line  per  square  centimeter 
in  a  path  1  cm.  long.  Since  the  product  of  field  intensity  and 
length  of  path  is  the  work  which  would  be  done  on  unit  pole  in 
moving  it  once  around  the  path,  unit  m.m.f.  will  produce  a 


ELECTROMAGNETISM  67 

field  intensity  of  such  a  value  that  1  erg  of  work  would  be 
done  in  moving  unit  pole  once  around  the  path.  The  longer 
the  path,  the  smaller  will  be  the  value  of  the  field  intensity. 
It  must  be  noted  that  m.m.f.  is  only  the  measure  of  the 
flux-producing  power  of  a  circuit  and  tells  nothing  as  to  the 
amount  of  flux  produced;  the  latter  depends  upon  the 
length  and  area  and  material  of  the  magnetic  circuit,  as  will 
be  shown  presently.  The  product,  NI,  is  frequently  called 
the  ampere-turns  of  the  circuit,  and  1  ampere-turn^  is  very 
commonly  used  as  a  unit  of  magnetomotive  force.  One  ampere- 
turn  is  evidently  equal  to  QAir  (or  1.257)  gilberts;  1  gil- 
bert of  m.m.f  is  produced  when  the  expression  OA^NI  is 
equal  to  unity;  that  is,  by  0.796  of  an  ampere-turn.  A 
given  number  of  ampere-turns,  NI,  does  not  require  that 
either  N  or  7  shall  have  any  particular  value,  but  that  the 
product  shall  have  the  specified  value.  That  is,  600  ampere- 
turns  will  be  given  by  2  amperes  through  300  turns  or  by  12 
amperes  through  50  turns;  and  the  flux-producing  power 
is  the  same  in  either  case. 

49.  Field  Intensity  and  Magnetizing  Force. — Let  Fig.  33 
represent  a  coil  of  wire  bent  around  so  that  its  ends  come 
together,  forming  a  uniformly  distributed  winding.  Let 
the  mean  length  of  the  axis  of  the  coil  be  I  centimeters  and 
let  the  intensity  of  the  field  along  the  axis  be  H;  then  the 
work  done  in  moving  a  unit  pole  once  around  the  path  is  HI] 
but  it  has  just  been  shown  that  this  work  is  also  equal  to 
therefore  the  value  of  the  field  intensity  is 

(45) 

The  term  "  magnetizing  force  "  is  frequently  used  to  desig- 
nate the  intensity  of  a  magnetic  field,  especially  in  connec- 
tion with  electromagnets.  Since  it  is  equal  to  the  ratio  of 
the  magnetomotive  force  to  the  length  of  the  path,  the  units 
very  commonly  used  are  "  gilberts  per  centimeter,"  "  am- 
pere-turns per  centimeter  "  or  "  ampere-turns  per  inch." 
A  field  intensity  of  1  line  per  square  centimeter  corre- 


68 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


spends  to  a  magnetizing  force  of  1  gilbert  per  centimeter. 
One  ampere-turn  per  centimeter  is  equal  to  1.257  gilberts 
per  centimeter  and  1  ampere-turn  per  inch  is  equal  to  0.495 
gilbert  per  centimeter. 

50.  Flux  Density;  Permeability. — When  the  magnetic 
circuit  consists  of  iron,  the  molecular  magnets  are  swung 
into  line  by  the  magnetizing  force  of  the  coil  and  the  lines 
of  force  due  to  these  are  added  to  the  lines  produced  by  the 
coil  alone;  that  is,  added  to  the  field  intensity  H.  If  the 
combined  strength  of  the  poles  of  the  molecular  magnets  is 


FIG.  33. 


equal  to  J  unit  poles  per  unit  of  cross-sectional  area,  then 
the  flux  per  unit  area  added  by  these  poles  will  be  4?rJ, 
since  there  are  4?r  lines  issuing  from  a  unit  pole.  The 
total  number  of  lines  of  force  per  square  centimeter  is  called 
the  density,  or  sometimes  the  induction,  and  is  generally 
represented  by  the  symbol  B  and  is  expressed  in  gausses. 
We  have,  then,  as  one  relation  between  B  and  H  that 
B  =  H  H-4irJ.  The  number  of  lines  per  square  centimeter 
added  by  the  presence  of  iron  depends  on  the  value  of  the 
magnetizing  force  QAirNI/l,  and  upon  the  composition 


ELECTROMAGNETISM  69 

of  the  iron.  In  any  case,  the  ratio  of  the  flux  density,  B, 
to  the  field  intensity,  H,  is  called  the  permeability  of  the 
iron  and  is  generally  represented  by  the  symbol  p.  We 
have  then  for  the  flux  density,  £,  in  a  magnetic  circuit  of 
permeability,  /*,  when  a  magnetomotive  force,  OAwNIj 
acts  upon  it,  the  equation, 

B-°-^.  '  (46) 

In  the  air,  the  density,  5,  is  numerically  equal  to  the  field 
intensity,  H  ,  and  the  permeability  of  air  is  taken  as  equal 
to  unity.  It  should  be  understood  clearly  that  H  =  QAirNI/l 
is  the  equation  for  field  intensity  or  magnetizing  force,  no 
matter  what  the  magnetic  circuit  may  consist  of,  and  it  is 
also  the  density  of  the  flux  when  the  magnetic  circuit  is  of 
air,  while  B  =  QAirNIiJi/l  is  the  equation  for  density  under 
all  conditions. 

51.  Total  Flux  Produced  by  a  Coil.—  The  total  flux,  </>, 
through  the  circuit  is,  of  course,  equal  to  the  product  of  the 
density,  £,  and  the  corresponding  cross-sectional  area,  A. 
That  is, 


(47) 


M.M.F. 
~~' 


This  equation  may  be  written, 


where  R  is  written  in  place  of  1/nA.  It  will  be  seen  that  this 
equation  is  entirely  similar  to  the  equation  for  current  in  an 
electric  circuit. 

52.  Reluctance.  —  The  quantity,  l/»A,  is  called  the 
reluctance  of  the  magnetic  circuit,  and  stands  in  the  same 
relation  to  the  magnetic  circuit  as  resistance  stands  to  the 
electric  circuit;  likewise,  flux  and  m.m.f.  in  the  magnetic 
circuit  have  the  same  relation  to  each  other  as  current  and 
e.m.f.  in  the  electric  circuit.  The  reciprocal  of  reluctance 
is  called  permeance  and  corresponds  to  conductance  in  the 


70 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


electric  circuit.  The  reluctance  of  a  cubic  centimeter  of  a 
substance  is  called  its  specific  reluctance,  or  its  reluctivity. 
Thus  the  reluctivity  of  air  is  unity.  It  should  be  noted 
that  permeability  is  a  ratio  and  that  it  corresponds  to  the 


5500 


5GOO 


4500 


3500 


^3000 


2500 


2000 


1500 


1000 


600 


Cast 


\ 


20  30  40  50   60  70   80  90  1JO  110  120  130 

Kilolincs  per  Square  Inch 

FIG.  34. — Permeability  Curves. 


reciprocal  of  relative  resistance  in  electric  circuits.  The 
permeability  of  iron  varies  with  the  density  and  therefore 
the  reluctivity  of  a  magnetic  circuit,  which  is  equal  to  unity 
divided  by  ju,  cannot  be  treated  as  a  constant  as  is  the 
specific  resistance  of  electric  circuits.  See  Fig.  34. 


ELECTRON  AGNETISM  71 

53.  Solution  of  Magnetic  Circuits. — In  most  cases,  the 
solution  of  magnetic  circuit  problems  relates  to  the  deter- 
mination of  the  ampere-turns  required  to  produce  a  given 
flux  density.  Usually,  a  certain  total  flux  is  required  and 
the  cross-sectional  area  of  the  circuit  is  made  such  that  the 
flux  density  will  not  exceed  the  limits  set  by  saturation  or 
hysteresis  and  eddy-current  losses.  It  should  be  noted 
that  the  number  of  ampere-turns  required  is  determined  by 
the  flux  density,  not  by  the  total  flux;  that  is,  a  large  flux 
will  be  produced  by  the  same  number  of  ampere-turns  as 
will  a  small  flux,  provided  the  ratio  of  total  flux  to  cross- 
sectional  area  is  not  changed.  It  should  also  be  noted  that 
the  flux  density  which  will  be  produced  in  an  iron  circuit 
by  a  given  number  of  ampere-turns  cannot  be  determined 
directly  because  the  permeability  varies  with  density  and 
there  is  no  practicable  mathematical  relation  between  them. 

Equation  (42)  may  be  writteri  in  the  form 

NI  =  Bl/QAir^  (49) 

and  the  ampere-turns  for  a  given  density  may  be  calculated 
from  this  formula  provided  data  are  at  hand  showing  the 
values  of  M  for  different  values  of  B  for  the  particular  kind 
of  iron  used.  Such  data  are  secured  from  test  pieces  of 
known  area  and  length,  the  flux  being  measured  for  various 
values  of  NI  from  zero  up  to  the  maximum  practicable 
value.  But  B  =  <j>/A  and  M  =  B/ H  =  Bl/QAirNI',  there- 
fore, corresponding  values  of  B  and  /*  can  readily  be  calcu- 
lated and  plotted  as  a  curve.  See  Fig.  34.  Or,  since  H 
^QAirNI/l,  the  curve  may  be  plotted  between  B  and  H. 
Or,  as  is  most  common  for  practical  purposes,  the  curve 
may  be  plotted  between  B  and  (NI/1);  such  curves  are 
called  Magnetization  Curves.  See  Fig.  35.  Typical  curves 
are  also  shown  in  the  Standard  Handbook  for  Electrical 
Engineers,  pp.  288-290.  Data  are  given  in  the  accom- 
panying table  for  various  kinds  of  iron,  from  which  the  mag- 
netization curves  may  be  plotted.  Using  such  curves,  to 
find  the  ampere-turns  required  for  any  density  in  any  length 


72 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


TABLE  I. — DATA  FOR  MAGNETIZATION  CURVES 
Ampere-turns  per  Inch. 


Square  Inch 
Density. 

Standard 
Sheet 
Steel. 

Wrought 
Iron. 

Silicon 
Sheet 
Steel. 

Soft 
Cast 
Steel. 

Spec. 
Alloy 
Steel. 

Cast 
Iron. 

10,000 

0.8 

2.4 

0.8 

3.1 

0.25 

7.0 

20,000 

1.4 

4.4 

1.4 

6.0 

0.40 

17.0 

30,000 

1.8 

6.2 

1.8 

8.8 

0.60           34.0 

35,000 

2.05 

7.0 

2.05 

10.1 

0.70 

40,000 

2.3 

7.7 

2.3 

11.6 

0.85 

45,000 

2.6 

8.4 

2.6 

13.2 

1.00 

50,000 

2.95 

9.0 

3.0 

14.9 

1.25 

55,000 

3.4 

9.6 

3.5 

16.7 

1.50 

60,000 

4.0 

10.4 

4.3 

18.8 

1.85 

65,000 

4.7 

11.4 

5.2 

21.4 

2.30 

70,000 

5.55 

12.9 

6.4 

24.8 

2.90 

72,500 

6.15 

13.8 

7.25 

27.0 

3.25 

75,000 

6.8 

14.9 

8.4 

29.5 

3.70 

77,500 

7.6 

16.2 

10.0 

32.7 

4.30 

80,000 

8.5 

17.7 

12.3 

36.2 

5.1 

82,000 

9.5 

19.2 

14.8 

39.4 

5.8 

84,000 

10.7 

21.0 

18.0 

43.0 

6.6 

86,000 

12.0 

23.5 

21.8 

47.0 

7.6 

88,000 

13.8 

26.4 

26.5 

52.0 

9.0 

90,000 

15.7 

30.0 

33.0 

58.0 

10.6 

92,000 

18.4 

34.1 

41.0 

64.0 

94,000 

21.6 

39.2 

51.0 

71.0 

96,000 

26.0 

47.0 

67.0 

80.0 

98,000 

32.6 

58.0 

87.0 

89.0 

100,000 

41.0 

70.0 

111.0 

100.0 

102,000 

52.0 

87.0 

140.0 

111.0 

104,000 

68.0 

107.0 

170.0 

125.0 

106,000 

88.0 

132.0 

200  0 

141.0 

108,000 

112.0 

162.0 

240.0 

160.0 

110,000 

138.0 

194.0 

280.0 

180.0 

112,000 

168.0 

225.0 

320.0 

208.0 

115,000 

222.0 

280.0 

400.0 

253.0 

120,000 

340.0 

395.0 

125,000 

500.0 

130,000 

700.0 

140,000 

1200.0 

150,000 

1700.0 

ELECTRON AGNETISM 


73 


of  iron,  it  is  only  necessary  to  multiply  the  given  length  of 
iron  by  the  value  of  (NI/l)  as  found  on  the  curve  for  the 
given  kind  of  iron  at  the  given  density. 


0    10    20    30  40    50   60    7J   80    90100110120130140150160.170180190200210220230 
Ampere-Turns  per  Inch 

FIG.  35.  —  Magnetization  Curves. 

54.  Series  Magnetic  Circuits.  —  When  a  series  magnetic 
circuit  is  made  up  of  different  kinds  of  iron  or  has  different 
densities  in  different  parts,  (NI)  is  found  for  each  part 
separately  and  the  total  (  NI)  is  the  sum  of  these. 

To  determine  the  ampere-turns  required  for  an  air  gap, 
it  is  only  necessary  to  solve  the  equation 

NI  =  Bl/OAir  =0.7.96£Z,  (50) 

where  B  is  in  gausses  and  /  is  in  centimeters.     If  B  is  in  ( 
lines  per  square  inch  and  I  is  in  inches,  the  formula  is 


(51) 


As  an  example  of  these  calculations,  consider  the  mag- 
netic circuit  shown  in  Fig.  36.  Suppose  the  piece  (c)  is  of 
sheet  steel  4  in.  long  and  has  a  cross-sectional  area  of  15 
sq.  in.;  pieces  (6),  (6)  are  each  of  wrought  iron  5  in.  long  and 


74 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


14  sq.  in.  in  cross-sectional  area;  piece  (d)  is  of  cast  steel, 
with  a  magnetic  path  45  ins.  long  and  20  sq.  ins.  in  cross- 
sectional  area;  and  the  two  air  gaps  (g),  (g)  are  each  0.08  in. 
long  and  14.5  sq.  ins.  in  cross-sectional  area.  Required  a 
flux  of  900,000  lines  through  the  circuit.  The  density  in 
piece  (c)  will  be  900,000/15=60,000  lines  per  square  inch; 
in  pieces  6,  6,  900,000/14=64,300  (results  carried  to  the 
third  significant  figure  only);  in  piece  (d),  45,000;  in  the  air 
gaps,  62,100.  Magnetization  curves  show  that  sheet  steel 
requires  4.0  ampere-turns  per  inch  at  60,000  lines  per  square 
inch;  that  wrought  iron  requires  11.3  at  64,300;  and  that 
cast  steel  requires  13.2  at  45,000.  The  two  air  gaps  will 


'  

—45^  
d                       Cast  Steel 

—  > 

<—  5- 
<  — 

^W.I.   „        1  |S.S. 

b      J3J75     / 

f    -W.I. 

<—  U  —  > 
3J5       , 

*      b 

9             9 
FIG.  36. 

require  (equation  47)  0.313x62,100x0.08x2=3100  am- 
pere-turns; piece  (c)  will  require  4.0x4  =  16  ampere- 
turns;  pieces  6,  b,  11.3x5x2  =  113  ampere-turns;  and 
piece  (d),  13.2x45=594  ampere-turns.  The  total  ampere- 
turns  required  are  then  3100  +  16  +  113+594=3823.  It 
should  be  particularly  noted  that  more  than  80  per  cent  of 
the  m.m.f.  is  used  in  carrying  the  flux  across  the  air  gaps. 
This  is  because  the  reluctance  of  air  is  so  much  greater  than 
that  of  iron.  Therefore  in  all  electrical  apparatus  where 
ah-  gaps  are  required  and  it  is  important  to  keep  the  ampere- 
turns  which  produce  the  flux  as  small  as  possible,  these  air 
gaps  are  made  as  small  as  is  practicable.  The  quantity, 
,  or,  BI/JJL,  for  any  part  of  a  magnetic  circuit,  is  fre- 


ELECTRON  AGNETISM 


75 


quently  called  the  drop  of  magnetic  potential  in  that  part 
of  the  circuit,  or  the  magnetic  potential  difference  between 
the  ends  of  that  portion  of  the  circuit.  This  is  evidently 
equal  to  the  m.m.f.  used  for  that  part  of  the  circuit  and  the 
sum  of  these  drops  taken  entirely  around  the  circuit  is  always 
equal  to  the  total  m.m.f.  acting  on  the  circuit. 

55.  Parallel  Magnetic  Circuits.— When  magnetic  cir- 
cuits are  in  parallel,  and  the  different  paths  have  the  same 
reluctance,  the  m.m.f.  calculated  for  one  of  the  paths  is  the 
m.m.f.  required  for  all  the  paths  in  parallel.  For  example, 
consider  Fig.  37.  Of  the  total  flux  in  the  middle  leg  (xy),  one 
half  will  pass  through  path  (a)  and  the  other  half  through 
path  (6).  The  densities  in  the  corresponding  parts  of  the 


CL  f     - 

-x<5 

i*l 

1 

Y 
0/2 

! 

• 

[    I 

*! 

i 

1 

1 

«fi 

j 

i 

V. 

i 

y 

m 


" 


FIG.  37. 


n 
FIG.  38. 


two  paths  will  be  the  same.  If  2000  ampere-turns  are 
required  to  produce  a  given  flux  in  path  (a),  the  same  2000 
ampere-turns  will  produce  an  equal  flux  in  the  path  (6),  if 
the  coil  is  placed  on  the  middle  leg.  However,  if  the  wind- 
ing is  placed  on  the  outside  legs,  2000  ampere-turns  will  have 
to  be  placed  on  each  of  these  legs.  If  the  density  in  the  mid- 
dle leg  is  the  same  as  in  the  outside  legs,  the  perimeter  of 
the  middle  leg  .will  be  greater  than  that  of  the  outside  legs; 
but  more  copper  will  be  required  to  produce  a  given  flux  in 
the  circuit,  if  coils  are  placed  on  the  outside  legs,  than  if  one 
coil  is  placed  on  the  middle  leg. 

Fig.  38  shows  the  corresponding  electric  circuit.  With 
a  constant  e.m.f.  between  (ra)  and  (m),  the  current  through 
path  (p)  will  be  the  same  whether  path  (q)  is  open  or  closed. 


76  ELECTRIC  AND  MAGNETIC  CIRCUITS 

No  more  e.m.f.  will  be  required  to  send  5  amperes  through 
(p)  and  5  amperes  through  (q)  than  will  be  required  to  send 
5  amperes  through  (p)  with  (q)  open,  if  the  current  density 
in  the  path  mBn  is  kept  the  same. 

56.  Size  of  Wire  Necessary  to  Produce  the  M.M.F. 
Required  for  a  Given  Magnetic  Circuit.  —  Let  it  be  supposed 
that  the  m.m.f.  has  been  calculated  for  a  given  magnetic 
circuit  and  (NI)  ampere-turns  are  found  to  be  required  for 
it.  If  E  is  the  voltage  to  be  applied  to  the  coil,  then  the 
resistance  of  the  coil  must  be 

R  =  E/I,  (52) 

where  /  is  the  current  which  the  coil  will  carry.  If  lt  is 
the  mean  length  (in  inches)  of  one  turn,  then  the  resistance 
must  also  be 

R  =  PNlt/12A,  (53) 

where  p  is  the  specific  resistance  of  the  wire  per  circular-mil- 
foot,  N  is  the  number  of  turns  of  wire  which  the  coil  will 
have  and  A  is  the  cross-sectional  area  of  the  wire  in  circular 
mils  (see  equation  11,  p.  39). 

Equating  these  two  expressions  for  R,  we  get 

A=PNIl,/12E.  (54) 

If  the  coil  is  of  copper  wire  and  the  running  temperature  be 
assumed  as  60°  C.,  the  value  of  p  will  be  12,  and  the  equa- 
tion becomes 

(55) 


The  area  of  the  wire  being  thus  determined,  the  current  will 
be  fixed  by  the  carrying  capacity  of  the  wire,  and  thence 
the  number  of  turns  may  be  found. 

57.  The  Field  Intensity  in  a  Solenoid.  —  From  the  pre- 
ceding discussion,  it  should  be  understood  that  the  m.m.f. 
of  a  coil  is  not  used  up  at  a  uniform  space  rate  around  the 
magnetic  circuit;  that  is,  the  m.m.f.  used  in  sending  the 
flux  through  1  in.  at  one  part  of  the  circuit  may  not  be  the 
same  as  that  used  in  1  in.  at  some  other  part  of  the  circuit. 


ELECTROMAGNETISM  77 

The  amount  used  in  any  given  portion  of  the  circuit  depends 
upon  the  reluctance  of  that  particular  portion,  just  as  the 
p.d.  between  different  points  of  an  electric  circuit  depends 
on  the  resistance  between  these  points.  The  reluctance  of 
any  part  of  an  air  circuit  is  equal  to  its  length  divided  by  its 
area;  if  the  reluctance  of  one  part  of  the  circuit  is  greater 
than  that  of  another  part,  the  former  part  will  require  the 
larger  part  of  the  m.m.f.  In  the  solenoid,  Fig.  39,  the 
cross-sectional  area,  Af,  of  the  path  within  the  coil  is  evi- 
dently very  small  as  compared  with  the  area,  A",  of  the 
return  path  outside  the  coil.  The  length  I",  of  the  part  of 
the  path  lying  outside  the  coil  is  indefinitely  greater  than 
Z',  the  part  within  the  coil,  but  still  the  reluctance  of  the 


F 


©©©©©©©©00000000GH" 

FIG.  39. 

entire  path  is  practically  all  within  the  solenoid.  On  account 
of  this,  the  flux  within  a  solenoid  may  be  taken  as  equal  to 
QAirNIA'  /I1  ',  where  V  is  the  length  of  the  solenoid  and  not 
the  length  of  the  whole  path.  That  is,  in  the  formula 


(56) 


the  reluctance  I"  /A"  is  negligibly  small  in  comparison  with 
V  I  A.'  and  may  be  neglected  when  V  is  large  compared  with 
Af.  This  relation  holds  when  lf  is  at  least  four  times  as 
great  as  the  mean  diameter  of  the  coil.  A  discussion  of  the 
case  for  short  solenoids  is  unnecessary  here  because  it  is 
seldom  necessary  in  practice  to  calculate  the  flux  in  such 
coils  except  when  dealing  with  alternating  currents.  The 
matter  will  be  taken  up  again  under  the  subject  of  induc- 
tance. 


78 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


58.  Magnetic  Leakage. — If  the  winding  which  produces 
the  flux  in  an  iron  magnetic  circuit  is  completely  distributed 
over  the  circuit  as  in  Fig.  40,  practically  all  the  flux  will  be 
confined  to  the  definite  path  in  the  iron  within  the  winding. 
But  if  the  winding  is  bunched  so  as  to  surround  only  a  por- 
tion of  the  iron  as  in  Fig.  41,  some  of  the  flux  will  take  paths 
through  the  surrounding  air.  If  there  is  a  gap  hi  the  iron 
circuit,  the  flux  will  not  pass  straight  across  the  gap  but  will 
spread  out  so  that  the  density  in  the  gap  will  be  less  than  in 
the  iron.  When,  for  some  definite  purpose,  a  specified 


/ 

/ 

\ 

f 

(  ( 

~ 





-N    N\ 
\ 

i  / 

1 

! 

'1 

! 

i 
i 

i 

! 

iiiy  IK! 

!  !  !  I!  1  11  ;  ! 

^- 

--^ 

- 

-^ 

FIG.  40. 


FIG.  41. 


flux  is  required  within  a  definite  air  gap  area,  the  lines  of 
force  which  do  not  so  pass  are  called  leakage  lines  and  the 
ratio  of  the  total  flux  produced  by  the  coil  to  the  useful 
flux  passing  across  the  definite  gap,  is  called  the  leakage 
coefficient.  This  coefficient  is  always  greater  than  unity. 
Its  calculation  from  the  dimensions  of  the  magnetic  circuit 
is  generally  difficult  and  complicated  and  it  is  usually  deter- 
mined by  actual  tests  on  different  forms  of  magnetic  circuit 
and  when  needed  that  coefficient  is  used  which  corresponds 
most  closely  to  the  type  of  circuit  used. 

59.  Hysteresis. — The  tendency  of  iron  to  retain  mag- 
netism is   called  hysteresis.     When   the   magnetizing  force 


ELECTRON  AGNETISM  79 

applied  to  an  iron  core  is  increased  the  flux  density  in  the 
iron  also  increases;  when  the  magnetizing  force  is  decreased, 
the  flux  density  likewise  decreases,  but  not  to  the  same 
value  it  had  with  the  same  magnetizing  force  when  increas- 
ing. To  magnetize  the  iron  to  a  certain  density  requires 
the  expenditure  of  energy;  when  the  magnetizing  force  is 
removed,  some  of  the  energy  is  returned  to  the  electric 
circuit,  but  not  all  of  it.  The  difference  between  the  energy 
put  into  the  magnetic  circuit  and  that  returned  to  the  elec- 
tric circuit  represents  the  tendency  of  iron  to  retain  the 
magnetism  and  is  called  hysteresis  loss.  When  an  iron  core 
is  magnetized  by  sending  current  through  a  coil  surround-' 
ing  the  core,  the  flux  through  the  core  will  increase  from  zero 
just  before  the  circuit  is  closed  to  a  certain  maximum 
after  the  circuit  is  closed.  The  value  of  the  final  flux  will 
depend  on  the  ampere-turns  of  the  coil  and  upon  the  reluc- 
tance of  the  core.  During  the  time  while  the  flux  is  growing 
an  e.m.f.  will  be  generated  in  the  coil,  the  value  of  which 
will  depend  on  the  rate  at  which  the  linkage  is  changing. 
If  there  are  N  turns  in  the  coil  and  the  flux  changes  by  an 
amount  d<i>  in  time  dt,  then  the  e.m.f.  will  be  W~8Nd(j)/dt 
in  volts,  and  if  the  current  at  a  given  instant  is  i  amperes, 
the  power,  or  rate  of  doing  work  will  be  W~8Nid(f>/dt  in 
watts.  By  Lenz's  Law,  which  we  have  already  discussed, 
this  e.m.f.  will  be  opposed  to  the  growth  of  the  current, 
and  the  work  done  in  forcing  the  current,  i,  through  the  cir- 
cuit against  this  e.m.f.  will  be  the  work  required  to  mag- 
netize the  core  by  the  amount  d<j).  If  dW  represents  the 
work  done  in  the  time  dt,  then  we  may  write 


dt  ~108(ft' 
or 

dW=Nid<j>/Ws.  (58) 

Now  the  flux  <j>  =  BA  and  d<f>  =  AdB,  where  B  is  the  density 
in  lines  per  square  centimeter  and  A  is  the  area  of  the  core 
in  square  centimeters;  also  Ni  =  Hl/QAir,  where  H  is  the 


80  ELECTRIC  AND  MAGNETIC  CIRCUITS 

field  intensity  and  I  is  the  length  of  the  core  in  centimeters. 
Therefore, 

JAJIdB_VHdB 
0.47rl08     0.47rlOs 

since  IA  is  the  volume,  V,  of  the  core.     Therefore  the  total 
work  done  in  magnetizing  a  core  up  to  a  density  B  is 


w  =»  HdB  j°uies- 


In  magnetizing  iron,  it  is  found  that  at  small  values  of  field 
intensity,  H,  the  increase  in  density  is  relatively  rapid  but 
as  H  becomes  larger  the  iron  approaches  a  saturated 
condition  and  the  density  increases  less  and  less  rapidly 
until  finally  the  presence  of  the  iron  adds  nothing  to  the 
flux  and  the  increase  in  density  becomes  equal  to  dH. 
There  is  no  known  mathematical  relation  between  B  and 
H  so  that  the  integral  of  HdB  cannot  be  mathematically 
determined.  If  the  iron  is  without  magnetism  at  the 
beginning,  the  manner  in  which  B  increases  wiihH  is  shown 
by  the  curve  Oa  in  Fig.  42.  The  area  Oayf  represents  the 
work  done  in  magnetizing  the  iron  to  a  value  of  B  repre- 
sented by  Oy1  ;  to  establish  this  fact,  consider  a  narrow  strip 
whose  width  is  dy  and  whose  length  is  Ox  when  the  ordinate 
is  Oy,  the  sum  of  all  such  strips  included  between  the  curve 
Oa  and  the  7/-axis  is  the  area  Oay';  but  dy  represents  a  cer- 
tain change  dB  in  the  flux  density  and  Ox  represents  the 
corresponding  value  of  H.  Therefore  xdy  represents  a 
certain  value  of  HdB  and  the  entire  area  Oay'  must  repre- 
sent the  integral  of  HdB  ;  that  is,  this  area  when  multiplied 
by  the  scales  used  for  H  and  for  B  and  by  the  constant, 
F/0.47rl08,  gives  the  work  done  in  magnetizing  the  volume  V 
to  a  density  B.  When  the  field  intensity  is  decreased,  the 
flux  linkages  decrease  and  again  an  e.m.f.  is  generated  which 
opposes  the  change;  that  is,  this  induced  e.m.f.  tends  to 
keep  the  current  from  decreasing  and  is  therefore  acting  in 
the  same  direction  through  the  electric  circuit  as  the  current. 


ELECTRON AGNETISM 


81 


It  is  therefore  giving  energy  back  to  the  electric  circuit,  and, 
in  amount,  it  is,  as  before,  equal  to  F/0.47rl08 )  HdB. 
However,  it  is  found  that  when  the  field  intensity  is  decreased, 
the  density  does  not  follow  the  same  curve  as  it  followed  on 
increasing  but  decreases  less  rapidly  and  when  H  has  been 
reduced  to  zero  the  iron  will  possess  a  certain  amount  of 
magnetism.  This  is  known  as  residual  magnetism  and  the 


o 


FIG.  42.  Hysteresis  Loop. 

density  corresponding  to  it  is  represented  by  Ob.  By  the 
same  reasoning  as  before,  the  area  bay'  represents  the 
energy  which  is  returned  to  the  circuit  when  the  field  inten- 
sity is  reduced  to  zero.  Therefore,  the  difference  between 
the  areas  Oay'  and  bay',  which  is  the  area  Oab,  represents 
the  energy  which  has  been  dissipated  in  the  process.  It 
is  found  when  iron  is  magnetized  and  then  demagnetized 
that  its  temperature  is  raised;  therefore,  heat  energy  must 
have  been  developed  in  the  iron,  and  this  heat  energy  is 


82  ELECTRIC  AND  MAGNETIC  CIRCUITS 

found  to  be  accounted  for  by  the  energy  represented  by  the 
area  Oab.  If  now  the  magnetizing  force  be  reversed,  work 
must  be  done  in  reducing  the  flux  to  zero  and  building  it 
up  to  a  value  y"  in  the  opposite  direction;  this  work  is 
represented  by  the  area  kgy".  When  H  is  again  reduced  to 
zero,  energy  represented  by  the  area  egy"  will  be  regained 
by  the  circuit,  and  if  H  be  reversed  again  and  increased 
to  Ox',  the  work  done  will  be  represented  by  the  area  eay'. 
The  total  energy  dissipated  by  a  complete  cycle,  acgfa,  is 
therefore  represented  by  the  area  included  between  the  two 
curves  acg  and  gfa.  The  phenomenon  which  causes  these 
two  curves  to  diverge  is  called  hysteresis  and  the  energy 
dissipated  when  a  piece  of  iron  is  magnetized  and  demag- 
netized is  called  hysteresis  loss.  It  is  sometimes  said  to  be 
due  to  molecular  friction.  Professor  Steinmetz  has  found 
that  when  a  piece  of  iron  is  subjected  to  repeated  reversals  of 
magnetism  from  a  given  maximum  density,  5,  in  one  direc- 
tion to  an  equal  maximum  value  in  the  opposite  direction, 
and  so  on,  the  power  lost  hi  hysteresis  can  be  represented 
by  the  formula 


P  (watte)-          -,  (61) 

where  /  is  the  number  of  complete  cycles  per  second,  V  is 
the  volume  (hi  cubic  centimeters)  of  iron  undergoing  the 
reversal,  B  is  the  value  of  the  maximum  density  (in  lines 
per  square  centimeter)  in  either  direction,  and  A;  is  a  con- 
stant depending  on  the  quality  of  the  iron.  If  V  and  B  are 
expressed  in  inch  units,  the  formula  is 


Pk  =fc/(16.38F)/107  =  0.83/c/T£L6/107.     (62) 

Fair  average  values  for  k  are  0.0012  for  good  annealed  sheet 
steel,  0.001  for  best  annealed  iron,  0.0008  for  good  silicon 
steel,  and  0.0006  for  best  silicon  steel.  For  typical  hys- 
teresis curves,  see  Standard  Handbook  for  Electrical  Engi- 
neers, pp.  290-292. 


ELECTROMAGNETISM  83 

60.  Eddy  Currents. — If  a  mass  of  iron  is  in  a  magnetic 
field  which  is  varying,  the  flux  will  cut  across  the  iron  in  a 
direction  at  right  angles  with  the  direction  of  the  flux,  and 
thus  generate  electromotive  forces  in  the  iron,  which  in  turn 
cause  currents  to  flow  in  it.  These  currents  are  called  eddy 
currents.  Their  energy  is  dissipated  in  heating  the  iron. 
Their  paths  are  more  or  less  indeterminate,  but  depend  in 
general  upon  the  shape  of  the  iron  with  respect  to  the  direc- 
tion of  the  flux.  The  energy  consumed  by  them  can  be 
greatly  reduced  by  laminating  the  iron  in  the  direction  of 
the  flux,  assuming  that  the  laminations  are  more  or  less 
completely  insulated  from  each  other  by  varnish  or  other 
insulating  material;  the  layer  of  oxide  formed  in  the  process 
of  annealing  is  often  sufficient  insulation.  When  a  given 
volume  of  iron  is  subjected  to  an  alternating  magnetic  flux, 
the  power  consumed  by  eddy  currents  can  be  represented 
by  the  formula,  P  =  KVt2f2B2,  where  K  is  a  constant  which 
depends  upon  the  conductivity  of  the  iron,  the  distribution 
of  the  flux,  the  manner  of  the  variation  of  flux  with  time, 
and  the  units  used;  V  is  the  volume  of  iron;  B  is  the  max- 
imum value  of  the  flux  density  in  each  direction;  /  is  the 
number  of  cycles  per  second;  and  t  is  the  thickness  of  the 
laminations.  The  above  equation  may  be  derived  as  fol- 
lows: Let  Fig.  43  represent  a  laminated  core  in  which  an 
alternating  flux  is  produced  by  the  coil  as  shown;  the 
direction  of  the  flux  will  be  perpendicular  to  the  paper. 
Let  I  represent  the  length  of  the  core  perpendicular  to  the 
paper,  w  the  width  of  the  laminations,  and  t  their  thickness. 
The  paths  of  the  eddy  currents  will  be  in  the  plane  of  the 
paper  as  shown  by  the  dotted  line  in  one  of  the  laminations. 
The  thickness  of  the  laminations  is  assumed  to  be  sufficiently 
small  with  respect  to  their  width  that  the  length  of  the  current 
path  can  be  represented  by  2w,  the  cross-sectional  area  of 
the  current  path  is  tl/2.  The  resistance  of  the  path  is  there- 
fore r=4Kiw/tl,  where  KI  is  the  specific  resistance  of  the 
iron.  The  total  flux  in  one  lamination  is  wtB,  and  the 
flux  cut  per  second  will  be  4fwtBj  since  the  flux  wtB  will  be 


84 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


cut  four  times  during  each  cycle.  The  effective  e.m.f.  will 
therefore  be  e  =  4K2fwtB,  where  K2  is  a  constant  depending 
on  the  manner  in  which  the  flux  varies  with  time.  The 
value  of  the  eddy  current  will  be  e/r  =  K2U2fB/Ki,  and 
the  power  loss  per  lamination  will  be  Pr,  or 


p  =  4(K22/Kl)wWf2B2.  (63), 

But  wit  is  equal  to  v,  the  volume  of  the  lamination,  so  that 
we  can  write  (substituting  K  for  4K22/  KI)J 

p  =  Kvt2B2/2.  (64) 


Coll 


FIG.  43. 

The  total  loss  will  be  pn,  where  n  is  the  number  of  lamina- 
tions; but  vn  is  the  total  volume  V  of  the  core,  so  that  the 
total  loss  is 

P  =  KVt2f2B2  (65) 

as  stated  above. 

This  formula  is  useful  as  showing  the  general  effect  of 
the  various  factors  on  the  eddy  current  loss,  but  it  is  not  very 
reliable  for  actual  calculations  on  account  of  the  fact  that 
certain  other  factors,  such  as  incomplete  insulation  between 
laminations,  variations  in  the  flux  distribution  and  its  man- 
ner of  change  with  time,  make  the  proper  selection  of  the 
constant  K  quite  uncertain.  In  practice  use  is  made  of 


ELECTROMAGNETISM 


85 


experimental  curves  secured  from  tests  under  the  desired 
conditions,  and  showing  total  core  loss  (hysteresis  and  eddy 
current  loss)  per  unit  volume  or  weight.  See  Standard 
Handbook  for  Electrical  Engineers,  pp.  291-292. 

The  usual  thickness  of  laminations  range  from  14  to 
28  mils  and  the  net  volume  of  iron  is  taken  as  from  85  to  95 
per  cent  of  the  stacked  volume. 

61.  Pull  of  an  Electromagnet. — It  has  been  proven 
(equation  60)  that  the  energy  stored  up  in  a  magnetic  field 
of  volume  V,  that  is,  of  length,  Z,  and  area  A,  is  equal  to 


TF  = 


HdB 


or 


HdB  ergs- 


FIG.  44. 

Now  the  energy  stored  in  any  particular  portion  of  the  circuit 
is  proportional  to  the  volume,  IA,  of  that  portion.  Suppose 
the  magnetic  circuit  consists  of  a  core,  C,  Fig.  44,  and  an 
armature,  D,  which  is  separated  from  the  core  by  gaps,  gg, 
each  of  area  A.  Let  the  length  of  the  gaps  be  increased  by 
an  infinitesimal  distance,  dx.  Then  the  energy  stored  in 
each  gap  will  be  increased  by  an  amount 


HdB. 


(67) 


86  ELECTRIC  AND  MAGNETIC  CIRCUITS 

But  in  the  gap,  the  permeability  is  unity,  and  B  =  H] 
therefore, 

r          B2Adx 

BdB=  (68) 


Adx  C 
=  —  I 


or 

dW    B2A 


But  since  dW  is  the  increase  in  energy  stored  in  the  gap  and 
is  proportional  to  dx,  it  represents  the  mechanical  work 
which  would  have  to  be  done  to  separate  the  armature 
from  the  core  by  the  additional  distance  dx;  therefore 
dW  /dx  is  the  average  value  of  the  force,  F,  required  to  pull 
the  armature  down  a  distance  dx.  This  force  is  in  dynes; 
expressed  in  pounds,  with  A  expressed  in  square  inches,  and 
B  in  lines  per  square  inch,  the  force  is 

T>2A 


72000" 

If  B  is  expressed  in  kilolines  per  square  inch  the  formula 
becomes 

F  =  0.01386£2A.  (706) 

62.  Inductance.  —  If  an  e.m.f.  of  120  volts  be  connected 
to  a  circuit  of  40  ohms  resistance  and  consisting  of  incan- 
descent lamps,  the  current  will  rise  almost  instantly  to  a  value 
of  3  amperes  as  may  be  observed  by  placing  an  ammeter 
in  the  circuit;  if,  however,  the  circuit  consists  of  a  coil  of 
wire  of  40  ohms  resistance  and  wound  upon  an  iron  core, 
the  current  may  require  several  seconds  to  obtain  its  final 
value  of  3  amperes.  Evidently  the  latter  circuit  possesses  to  a 
much  greater  degree  than  the  former  some  property  which 
opposes  the  change  in  current.  It  shows  that  electricity, 
like  matter,  resists  a  change  in  its  state  of  motion  and  thus 
has  a  property  similar  to  that  which  we  call  inertia.  Men- 
tion has  already  been  made  of  the  manner  in  which  this 
opposition  manifests  itself,  namely  the  generation  of  a  back 
e.m.f.  by  the  changing  flux  which  is  linked  with  the  electric 


ELECTRON  AGNETISM  87 

circuit.  This  property  of  an  electric  circuit,  by  virtue  of 
which  an  opposing  e.m.f.  is  generated  whenever  an  attempt 
is  made  to  change  the  current  in  it,  is  called  self-induction. 
When  the  current  in  the  wire  is  increased  or  decreased  the 
magnetic  flux  surrounding  the  wire  increases  or  decreases 
with  it;  the  lines  of  force  seem  to  expand  out  from  the 
center  of  the  wire  when  the  current  increases  and  to  collapse 
toward  the  center  of  the  wire  when  the  current  decreases. 
Thus  they  cut  across  the  wire  and  generate  an  e.m.f.  and 
this  e.m.f.  is  always  in  such  a  direction  as  to  oppose  the 
change  in  current  which  produces  it.  The  magnitude  of 
the  opposing  e.m.f.  depends,  of  course,  upon  the  rate  at 
which  the  flux  linkage  is  changing,  which,  in  turn,  depends 
upon  the  value  of  the  current  and  the  rate  at  which  it  is 
changing.  Let  es  represent  the  e.m.f.  of  self-induction;  it 
is  equal  to  Nd^/lWdt,  where  N  is  the  number  of  turns  in 
the  circuit  and  <f>  is  the  flux  which  links  it.  But  it  has  been 
shown  that  ^=^A^^ANi/l]  therefore,  d^^OAi 
substituting  this  value  of  d<{>,  we  get 


e-<r-T&rtr 

'0.47ruA  N2\ 
The  quantity  ( T^, — )  is  called  the  inductance  of  the 


circuit,  and  is  represented  by  the  symbol  L.  Note  that 
when  there  is  iron  in  the  magnetic  circuit,  the  permeability 
varies  with  the  flux  and  L  is  not  constant ;  when  the  magnetic 
circuit  consists  of  air  and  its  length  and  area  are  perfectly 
definite,  L  is  evidently  a  constant  and  easily  calculated,  but 
magnetic  circuits  of  ah*  are  usually  not  very  definite  in 
length  or  area.  The  calculation  of  L  is  therefore  generally 
not  a  simple  matter.  There  are  several  ways  in  which 
inductance  is  defined.  Equation  (71)  may  be  written, 
es  =  Ldi/dt,  from  which  we  get 


(72) 


88  ELECTRIC  AND  MAGNETIC  CIRCUITS 

and  from  this  equation  L  is  defined  as  the  ratio  of  the  e.m.f. 
induced  in  a  circuit  to  the  rate  of  change  of  current  which 
induces  it.  This  should  be  taken  as  the  most  fundamental 
definition  since  it  connects  most  closely  with  the  property 
of  self-induction  as  defined  above.  The  practical  unit  of 
inductance  is  one  henry.  A  circuit  has  an  inductance  of 
one  henry  when  current  in  it  changing  at  the  rate  of  1  ampere 
per  second  causes  an  e.m.f.  of  1  volt  to  be  generated.  If  a 
current  changing  at  the  rate  of  1  ampere  per  second  causes 
an  e.m.f.  of  5  volts,  the  inductance  is  5  henries.  A  henry 
is  a  larger  unit  than  is  generally  met  in  practice,  and  a 
smaller  unit,  the  millihenry  (1/1000  of  a  henry)  is  much  used. 
If  es  is  expressed  in  abvolts  and  i  in  abamperes,  L  is  given 
in  abhenries;  the  abhenry  is  therefore  equal  to  1  henry 
divided  by  109,  since  1  volt  is  108  abvolts  and  1  ampere  is 
1/10  abampere. 

The  factors  which  go  to  make  up  the  values  of  L  are,  of 


.     ..  .  , 

course,  those  in  the  expression  I  --  j^  --  ),  from  which  it 

should  be  noted  that  L  varies  directly  as  N2,  A,  and  /*,  and 
inversely  as  the  length  I.  This  expression,  however,  will 
not  give  correct  values  for  L  except  in  the  case  of  an  air- 
cored  coil  closed  on  itself  as  shown  in  Fig.  33  and  in  which 
the  cross-sectional  area,  A,  is  uniform  and  I  is  the  mean 
length  of  the  magnetic  path  inside  the  coil.  It  is  approxi- 
mately correct  for  a  long  solenoid. 

Inductance  is  also  defined  as  the  rate  of  change  of  flux- 
linkages  with  current.  In  abvolts,  es=Nd^>/dt]  if  this 
expression  is  substituted  for  es  in  equation  (72)  we  get 


T  ,7  7     ... 

L=  =Nd<f>/di,  (73) 


which  is  the  mathematical  expression  for  the  definition  just 
given.  In  this  expression,  i  is  in  abamperes  and  L  is  in 
abhenries. 

In  the  case  of  an  air  circuit,  d<f>/di  is  constant  and  equal 
to  4>/ij  and  L  may  be  expressed  as  N<t>/i,  or  defined  as  flux- 


ELECTROMAGNETISM 


89 


linkages  per  unit  current.  This  expression  is  the  most 
convenient  one  for  calculating  the  inductance  of  a  trans- 
mission line  as  will  be  shown  later. 

For  calculating  the  inductance  of  circular  air-cored  .coils 
of  any  length,  depth  and  diameter,  Professor  Brooks,  of 
the  University  of  Illinois,  has  worked  out  an  empirical  for- 
mula which  will  give  results  correct  within  1  per  cent.  The 
formula  is 

25.07       N2(c+d)2 


;     (74) 


FIG.  45. 

in  henries,  where  N  is  the  number  of  turns  in  the  coil  and 
b,  c,  and  d,  are  the  dimensions,  in  inches,  shown  in  Fig.  45. 
It  was  found  that  the  maximum  inductance  for  a  given 
number  of  feet  of  wire  would  be  given  when  b=Q.6d  and 
c=0.5d  Substituting  these  values  in  the  above  formula, 
we  get 

L=  (75) 


90  ELECTRIC  AND  MAGNETIC  CIRCUITS 

as  the  inductance  of  a  coil  when  wound  in  such  shape  as  to 
give  the  maximum  value. 

63.  Growth  of  Current  in  an  Inductive  Circuit.—  When 
an  e.m.f.  E  is  applied  to  a  circuit  containing  a  resistance  r 
and  an  inductance  L,  the  current  will  not  instantly  assume 
the  value  E/r,  but  will  rise  at  a  rate  depending  on  the  ratio 
of  L  to  r.  With  large  L  and  small  r,  it  will  grow  slowly, 
while  with  small  L  and  larger  r,  it  will  grow  rapidly.  At  any 
instant  during  its  growth,  the  e.m.f.  used  in  overcoming  the 
resistance  will  be  equal  to  n,  while  the  remainder  will  be 
used  in  overcoming  the  e.m.f.  of  self-induction,  which  is 

di 
equal  to  L-r'  therefore,  we  can  write  the  equation  for  the 

total  e.m.f. 

di 

(76) 


To  solve  this  equation  for  i,  it  must  be  arranged  in  a  form 
which  can  be  integrated.     To  do  this,  write 

(E-ri)dt  =  Ldi  (77) 

dt       di 


Then  integrating, 

£=-ilog,(#-n)  +  K  (79) 

The  value  of  K  is  found  by  the  condition  that  when  t=0, 
i=0',  whence 

K=^\ogeE  (80) 

and 

-g-**.  (^),  '          (8.) 

or 

e"rt/L  =  T'  (82) 

whence, 

i  =  1(1-6-"*),  (83) 


ELECTROMAGNETISM 


91 


from  which  equation  the  value  of  the  current  can  be  cal- 
culated for  any  tune,  t,  after  the  circuit  is  closed.  When  a 
sufficient  time  has  elapsed  that  e~rt/L  becomes  sensibiy 
equal  to  zero,  then  i  =  E/r. 

The  curve  for  equation  (83)  is  shown  in  Fig.  46.  The 
ratio  r/L  is  generally  so  large  that  only  a  fraction  of  a 
second  is  required  to  meet  this  condition.  When  t  =  L/r, 
then  rt/L  =  l  and 

-          «      (84) 


The  ratio  L/r  is  called  the  time  constant  of  the  circuit  and 
is  the  time  required  for  the  current  to  reach  63.2  per  cent  of 
its  final  steadv  value. 


FIG.  46. — Growth  and  Decay  of  Current  in  an  Inductive  Circuit. 

64.  Decay  of  Current  in  an  Inductive  Circuit. — When  a 
coil  is  short-circuited  and  the  source  of  e.m.f.  removed,  a 
current  will  be  found  to  continue  to  flow  for  a  short  time  in 
the  same  direction  as  formerly  and  it  can  only  be  accounted 
for  by  the  e.m.f.  which  is  generated  in  the  coil  by  the  decreas- 
ing magnetic  field.  Since  the  impressed  e.m.f.  on  the  cir- 
cuit is  now  zero,  the  equation  for  the  circuit  is 


0 


or 


rdt 
L 


(85) 


(86) 


92  ELECTRIC  AND  MAGNETIC  CIRCUITS 

Integrating,  we  get 

/r/\ 

(87) 


Let  i'o  represent  the  steady  value  (E/r)  which  the  current 
had  at  the  instant  the  circuit  was  short-circuited.  Then 
the  constant  of  integration,  K,  is  found  from  the  condition 
that  when  t=0,  i  =  /o;  or, 

JT=-log,70,  (88) 

therefore, 

(89) 

(90) 
(91) 

/ 


*      /      /      / 


FIG.  47. 

From  this  it  is  seen  the  current  dies  away  according  to  the 
same  law  by  which  it  increases  when  the  circuit  is  first  closed. 
The  curve  for  equation  (91)  is  shown  in  Fig.  46. 

A  circuit  which  has  a  sensibly  large  time  constant, 
that  is,  one  in  which  the  current  grows  at  a  relatively  slow 
rate,  is  called  an  inductive  circuit.  When  an  inductive 
circuit  is  opened  quickly  the  magnetic  field  associated  with 
it  dies  away  very  rapidly  and  may  generate  a  very  large 
e.m.f.  The  spark  which  occurs  where  such  a  circuit  is 
opened  is  due  to  this  e.m.f.  and  if  the  circuit  be  opened  too 
quickly,  the  e.m.f.  may  be  great  enough  to  puncture  the 
insulation  of  the  circuit  and  cause  considerable  damage. 

It  is  impossible  for  a  circuit  to  be  perfectly  non-inductive, 
but  if  it  is  desired  to  wrind  a  quantity  of  wire  into  a  coil 
so  that  it  shall  be  practically  non-inductive,  the  wire  may 
be  wound  back  on  itself  as  illustrated  in  Fig.  47.  That  is, 
two  wires  are  wound  side  by  side,  their  inside  ends  being 
connected  together,  and  their  outside  ends  forming  the 


ELECTRON  AGNETISM  93 

terminals  of  the  coil.  The  current  then  flows  in  one  direc- 
tion around  the  core  through  half  the  turns,  and  in  the  oppo- 
site direction  through  the  other  half,  so  that  the  m.m.f. 
of  one  half  is  opposed  to  that  of  the  other  half  and  no  flux 
is  produced  except  what  little  may  pass  between  adjacent 
wires.  The  coil  is  therefore  practically  non-inductive. 
Another  method  of  making  a  non-inductive  resistance  is  to 
wind  the  wire  on  a  very  thin  flat  card  so  that  the  area 
included  within  the  turns  is  so  small  that  an  inappreciable 
amount  of  flux  is  linked  with  them. 

65.  Energy  of  a  Magnetic  Field. — Whenever  a  current 
flows  against  the  resistance  of  a  wire  or  against  a  counter- 
e.m.f.,  electric  power  is  consumed.  The  power  consumed  in 
overcoming  resistance  is  n'2,  the  power  required  to  overcome 
a  counter-e.m.f.  is  equal  to  the  product  of  the  current  and 
the  counter-e.m.f.  and  the  total  power  is  equal  to  the  sum 
of  these,  or  to  the  product  of  the  current  and  the  total  im- 
pressed e.m.f.  Thus,  in  the  case  of  an  inductive  circuit  on 
which  an  e.m.f.  E  is  impressed, 

fli 

Ei=ri2+Li^t.  (92) 

It  is  evident  that  the  power,  Li—,  will  become  zero  as  soon 

as  the  current  reaches  its  steady  value,  and  all  the  power 
supplied  will  be  used  in  overcoming  the  resistance.  When  a 
steady  current  is  flowing  in  a  circuit,  the  magnetic  field 
linked  with  it  is  steady  and  repeated  experiments  have 
proven  that  a  magnetic  field  requires  no  energy  to  maintain 
it;  it  only  requires  energy  to  establish  it  or  to  increase  it, 
and  it  returns  this  energy  to  the  electric  circuit  when  it 
decreases. 

The  work  done,  or  the  energy  transformed,  during  any 
period,  dt,  will  be, 

Eidt  =  ri2dt + Lidi.  (93) 

That  part  of  the  energy  represented  by  Lidi  is  spent  in  build- 
ing up  the  magnetic  field.  This  point  is  sometimes  rather 


94  ELECTRIC  AND  MAGNETIC  CIRCUITS 

difficult  to  comprehend,  but  it  should  be  remembered  that 
the  opposition  to  the  rising  current  comes  from  the  fact  that 
the  increasing  magnetic  field  generates  the  counter-e.m.f. 
and  the  counter-e.m.f.  disappears  as  soon  as  the  magnetic 
field  becomes  steady,  so  that  the  energy  which  was  used 
must  now  be  located  hi  the  magnetic  field.  If  the  last 
equation  be  integrated  between  t=0  and  any  later  time  T, 
we  get 

faTEidt=f0Tri*dt+iLP,  (94) 

where  7  is  the  value  of  the  current  at  time  T.  The  last  term 
in  this  equation  represents  the  energy,  in  joules,  which  is 
stored  up  in  a  magnetic  field  when  the  current  which  is  pro- 
ducing it  is  /  amperes  and  the  inductance  of  the  circuit  is 
L  henries.  An  idea  of  the  amount  of  energy  associated  with 
a  magnetic  field  may  be  obtained  from  an  example.  Sup- 
pose 2000  turns  of  wire  be  wound  into  a  coil  of  10  ins.  inside 
diameter  so  as  to  give  the  maximum  inductance;  by  the 
empirical  formula  (75)  given  above  the  inductance  will  be 

_     30.64X2000^X10 

L  =  -  -.Q9  -  =  1.226  henries. 

If  a  current  of  20  amperes  flows  in  the  coil,  the  energy  of  the 
field  will  be 

0.5X1.226X400=245  joules, 

or  245/1.356  =  181  ft.-lbs.,  that  is,  enough  energy  to  raise 
181  Ibs.  a  distance  of  1  ft.  After  the  current  becomes  steady, 
the  difference  between  the  total  energy  which  has  been 
supplied  and  the  energy  which  has  been  dissipated  in  heating 
the  circuit  is  found  to  be  constant  and  equal  to  %LP,  thus 
showing  that  no  energy  is  required  to  maintain  the  magnetic 
field.  This  may  also  be  shown  by  measuring  the  power  at 
any  instant;  all  of  the  power  supplied  can  be  accounted  for 
by  the  rate  at  which  heat  is  dissipated  in  the  coil. 

66.  Inductance  of  Two  Long  Parallel  Wires. — An  im- 
portant factor  in  the  design  and  operation  of  transmission 
lines  is  the  inductance  of  such  lines.  The  problem  is  to  find 


ELECTRON  AGNETISM 


the  flux  per  unit  current  linking  the  circuit  made  up  of  two 
long  parallel  wires.  Evidently,  if  we  find  this  for  unit 
length  of  circuit,  we  can  calculate  it  for  any  length  of  circuit 
by  simple  multiplication.  Let  Fig.  48  represent  two  par- 
allel wires,  A  and  B,  of  radius  r  and  distance  D  apart  from 
center  to  center.  We  have  seen  (Article  45)  that  the  field 
intensity,  H,  at  a  distance  x  from  the  center  of  a  wire 
carrying  7  abamperes  of  current  is  21  /x.  The  total  flux 
per  centimeter  length  of  wire  A  due  to  the  current  in  wire  A 
and  passing  between  the  surface  of  wire  A  and  the  center 
of  wire  B  is  therefore 


D2Idx  D 

=  2/loge  — . 


x 


(95) 


FIG.  48. 


FIG.  49. 


We  have  yet  to  find  the  flux  inside  the  wire  A  and  add  it  to 
4>'.  Let  Fig.  49  be  an  enlarged  cross-section  of  wire  A. 
Assume  the  current  I  to  be  uniformly  distributed  over  the 
cross-section  of  the  wire;  then  the  current  within  the  radius 
x  is  If=x2I/r2.  The  flux  density  at  distance  x  from  the 
center  is  21'  /x  and  the  flux  in  the  ring  of  width  dx  and  length 
1  cm.  perpendicular  to  the  paper  is  2I'dx/x  or  2Ixdx/r2. 
This  flux  links  only  with  the  current  /'  and  the  value  of  the 
flux  which  would  link  the  current  7  is  smaller  than  this  in 
the  ratio  x2/r2  or  is  equal  to  2Ix3dx/r4.  Integrating  this 
between  0  and  r,  and  we  have  the  total  equivalent  flux 

within  the  wire  as 

7 


96  ELECTRIC  AND  MAGNETIC  CIRCUITS 

The  total  flux  then  linking  each  wire  per  centimeter   of 
length  is  0'  +  </>",  or 


,  (97) 

and  the  flux  per  unit  current,  or  the  inductance,  is 

)+i  (98) 

in  abhenries  per  centimeter,  or, 
L  =  2.54  X  12  X5280  X  (2  X2.303  logy  +  O.s)l0-9 

=  (o.74  log  y+  0.0805^  10-3,     (99) 

in  henries  per  mile  of  wire,  using  common  logs  instead  of 
Napierian  logs.  Since  there  are  103  millihenries  in  1  henry, 
the  inductance  in  millihenries  per  mile  is 

L  =  0.74  log  y  +0.0805.  (100) 

In  millihenries  per  1000  ft.  it  is 

L  =0.14  logy  +0.015.  (101) 

67.  Skin  Effect.  —  In  Article  66,  it  was  assumed,  in  arriv- 
ing at  the  amount  of  flux  within  the  wire,  that  the  current 
was  uniformly  distributed  throughout  the  wire.  With 
varying  or  alternating  current,  the  distribution  will  not  be 
uniform.  A  solid  wire  may  be  considered  as  made  up  of  a 
very  small  cylindrical  core  surrounded  by  very  thin  cylin- 
drical shells  of  increasing  diameter,  and  each  fitting  tightly 
over  the  one  of  next  smaller  diameter.  The  flux  surrounding 
(that  is,  linking  with)  any  shell  is  the  flux  produced  by  that 
shell  and  all  the  shells  outside  of  it,  but  not  including  the 
flux  produced  by  the  shells  inside  of  it;  therefore  the  core 
is  linked  with  more  flux  than  the  shell  next  to  it,  and  each 
shell  is  linked  with  more  flux  than  the  next  shell  outside  of  it. 


ELECTRON  AGNETISM  97 

The  result  is  that  the  inductance  at  the  center  of  the  wire 
is  greater  than  at  the  surface,  and  the  back-e.m.f.  of  self- 
induction  is  greater  at  the  center  and  the  current  at  the 
center  of  the  wire  is  less  than  at  the  surface.  The  current 
may  be  thought  of  as  crowded  toward  the  surface  of  the 
wire;  this  phenomenon  is  known  as  skin  effect.  It  has  the 
apparent  effect  of  increasing  the  resistance  of  the  wire, 
because  less  current  flows  in  the  wire  than  would  flow  if 
it  were  a  steady  direct  current.  The  deviation  from  uni- 
formity of  current  distribution  depends  upon  the  size  of  the 
wrire,  the  rapidity  with  which  the  current  is  changing,  and  the 
material  of  which  the  wire  is  made.  With  60-cycle  alter- 
nating current  and  copper  or  other  non-magnetic  wires  of 
less  than  400,000  circular  mils  area,  the  apparent  resistance 
is  less  than  1  per  cent  greater  than  the  direct  current  resist- 
ance. With  iron  or  steel  wire,  however,  the  apparent 
resistance  may  be  several  times  greater,  even  with  wires 
as  small  as  No.  6  B.  &  S.,  owing  to  the  greater  permeability 
of  those  materials. 

68.  Mutual  Induction. — When  two  coils  are  so  placed 
with  reference  to  each  other  that  the  flux  which  links  one 
coil  will  also  link  the  other,  either  wholly  or  in  part,  then  if 
the  current  in  one,  which  we  may  call  the  primary,  varies, 
the  varying  flux  will  not  only  produce  an  e.m.f.  of  self- 
induction  in  the  primary  coil,  but  will  produce  an  e.m.f. 
of  mutual  induction  in  the  other,  or  secondary  coil.  This 
is  the  principle  of  the  alternating  current  transformer. 
The  value  of  the  e.m.f.  induced  in  the  secondary  coil  depends 
on  how  completely  the  flux  produced  by  the  primary  links 
the  turns  of  the  secondary,  and  on  the  number  of  turns  in 
the  secondary.  When  the  two  coils  are  wound  close  together 
on  an  iron  core,  then  the  whole  flux  will  very  nearly  com- 
pletely link  both  coils.  If  the  number  of  turns  in  the  pri- 
mary coil  be  N'  and  in  the  secondary  coil  be  N",  and  the 
flux  be  changing  at  the  rate  d$/dt,  then  the  e.m.f.  induced 
in  the  primary  will  be  Nfd$/lQPdt  and  in  the  secondary  it 
will  be  N"d$/lQPdt]  that  is,  the  ratio  of  the  primary  to 


98  ELECTRIC  AND  MAGNETIC  CIRCUITS 

secondary  induced  e.m.f's  will  be  equal  to  the  ratio  of 
primary  to  secondary  turns. 

The  ratio  of  the  e.m.f.  induced  in  circuit  No.  2  to  the 
rate  of  change  of  current  in  circuit  No.  1  is  called  the  co- 
efficient of  mutual  induction.  This  coefficient  depends  upon 
the  geometrical  arrangement  of  the  circuits  and  the  per- 
meability of  the  magnetic  material  surrounding  them.  If 
the  permeability  remains  constant,  the  coefficient  of  mutual 
induction  of  circuit  No.  1  with  respect  to  circuit  No.  2 
is  the  same  as  that  of  circuit  No.  2  with  respect  to  circuit 
No.  1.  Formulae  for  calculating  this  coefficient  are  given 
hi  the  Standard  Handbook  for  Electrical  Engineers,  p.  74. 

"  Cross-talk  "  between  parallel  telephone  lines  and  dis- 
turbances in  telephone  lines  which  are  parallel  to  power 
transmission  lines  are  frequently  caused  by  mutual  induc- 
tion. These  may  be  reduced  and  sometimes  practically 
eliminated  by  proper  "  transposition."  For  methods  of 
transposition,  see  Standard  Handbook  for  Electrical  Engi- 
neers, p.  1678. 


CHAPTER  V 


ELECTROSTATICS 

69.  Electric  Charges. — Take  two  pieces  of  metal  A  and 
B  (that  is,  two  conductors)  and  let  them  be  insulated  from 
each  other  and  from  all  other  conductors.  They  may  be 
two  wires  of  a  transmission  line  or  two  pieces  of  tinfoil 
separated  by  a  sheet  of  paper  or  a  sheet  of  mica.  Let  a 
galvanometer  be  connected  to  each  piece  as  shown  in  Fig.  50, 
each  galvanometer  being  so  connected  that  if  current 


FIG.  50. 

flows  toward  the  right,  the  galvanometer  needle  will  be 
deflected  toward  the  right.  Now  let  side  A  be  connected 
to  the  positive  terminal  of  a  source  of  electromotive  force 
and  side  B  to  the  negative  terminal  as  shown.  At  the 
instant  the  connection  is  made  the  needle  of  galvanometer  C 
will  be  deflected  to  the  right  while  that  of  D  will  be  deflected 
to  the  left;  both  needles  will  then  come  back  to  rest  at 
zero.  The  two  wires  are  now  said  to  be  charged  with  elec- 
tricity. Electricity  is  said  to  have  flowed  into  wire  A  and 
charged  it  positively;  electricity  is  said  to  have  flowed  out 
of  wire  B  and  charged  it  negatively.  Every  point  on  A  is 

99 


100  ELECTRIC  AND  MAGNETIC  CIRCUITS 

now  at  the  same  potential  as  the  positive  terminal  of  the 
generator,  and  every  point  on  B  is  at  the  same  potential 
as  the  negative  terminal  of  the  generator;  from  Ohm's 
Law,  we  have  learned  that  there  is  no  potential  difference 
between  two  points  in  the  same  conductor  unless  there  is  a 
current  flowing.  The  difference  of  potential  between  A 
and  B  is  equal  to  the  electromotive  force  of  the  generator. 
The  circuit  may  now  be  opened  and  closed  at  will  at  E  and 
F  and  no  effect  will  be  seen  on  the  galvanometers.  Assum- 
ing perfect  insulation  between  the  two  wires,  they  will 
remain  charged.  That  they  do  remain  charged  after  being 
disconnected  from  the  source  of  e.m.f.  may  be  seen  by  dis- 
connecting them  from  the  generator  at  E  and  F  and  con- 
necting GtoH',  the  needle  of  C  will  be  deflected  to  the  left 
and  that  of  D  to  the  right;  that  is,  electricity  will  flow 
out  of  A  into  B  until  the  two  are  at  the  same  potential. 

70.  The  Electrostatic  Field. — Let  the  wires  A  and  B 
again  be  charged  as  before;  with  suitable  apparatus,  it 
would  be  found  that  there  exists  between  them  a  force  of 
attraction.  This  is  known  as  electrostatic  attraction  and 
it  shows  that  the  region  surrounding  and  between  two  bodies 
charged  to  different  potentials  is  in  a  state  of  stress.  Such 
a  region  is  called  an  electrostatic  field  and  the  material  sep- 
arating two  charged  bodies  is  called  a  dielectric.  It  is 
found  by  experiment  that  when  the  dielectric  is  air  (or 
vacuum)  the  magnitude  of  the  stress,  or  the  intensity  of 
the  electrostatic  field,  depends  directly  upon  the  potential 
difference  between  the  conductors,  and  inversely  upon  the 
distance  between  them. 

As  electrostatic  field  can  be  conveniently  pictured  by 
imagining  lines  to  be  drawn  from  one  of  the  charged  bodies 
to  the  other;  at  points  where  the  force  is  strong  the  lines 
should  be  thought  of  as  crowded  together  and  at  points 
where  the  force  is  weak  the  lines  should  be  thought  of  as 
spread  apart.  Fig.  51  illustrates  the  distribution  of  these 
lines  in  every  plane  at  right  angles  to  two  equally  charged 
straight  parallel  wires.  The  actual  distribution  will  of 


ELECTROSTATICS 


course  vary  with  the  distance  between  the  wires  and  the 
ratio  of  the  diameter  of  the  wires  to  the  distance  between 
their  centers.  The  same  figure  also  illustrates  the  distri- 
bution of  the  lines  in  every  plane  through  the  centers  of  two 
equally  charged  spheres.  In  the  case  of  parallel  conducting 
plates  which  are  large  in  surface  area  relative  to  their  dis- 
tance apart,  the  lines  will  be  uniformly  distributed  in  the 
space  between  the  plates,  except  near  the  edges.  See  Fig.  52. 
These  lines  are  called  electrostatic  lines  of  force.  The  meas- 


FIG.  51. 


ure  of  the  intensity  of  an  electrostatic  field  at  any  point  is 
taken  as  the  force  which  would  be  exerted  by  it  on  a  unit 
quantity  of  electricity  placed  at  the  given  point  when  the 
dielectric  is  air.  In  the  original  development  of  the  theory 
of  electrostatics,  the  unit  of  quantity  was  taken  to  be  that 
quantity  which  when  placed  1  cm.  from  a  similar  and  equal 
quantity  would  repel  it  with  a  force  of  1  dyne.  This  unit 
is  called  the  electrostatic  unit.  An  electrostatic  field  then 
has  an  intensity  of  one  line  per  square  centimeter  when  it 
exerts  a  force  of  1  dyne  on  an  electrostatic  unit  of  quantity. 
Therefore,  since  there  are  4?r  square  centimeters  of  area  in 


ELECTRIC  AND  MAGNETIC  CIRCUITS 

the  sphere  of  unit  radius  surrounding  a  quantity  there 
will  be  4?r  lines  of  force  issuing  from  unit  quantity,  or 
charge. 

When  the  dielectric  is  other  than  air,  it  is  found  that  the 
charge  which  a  given  potential  difference  will  produce  on 
the  conductors  is  increased  and  consequently  the  lines  of 
force  are  increased  in  number.  The  sum  of  the  original 
intensity  (in  air)  and  the  added  lines  per  square  centimeter 
is  called  the  flux  density,  and  the  ratio  of  the  density  to  the 
intensity  is  known  as  the  dielectric  constant  of  the  dielectric. 
It  is  also  called  specific  inductive  capacity. 

Since  the  lines  of  force  are  proportional  to  the  amount 
of  charge  on  the  conductors  separating  the  dielectric,  this 
constant  may  be  determined  by  measuring  the  charge  (by 
means  of  a  ballistic  galvanometer)  on  the  conductors  with 
the  given  dielectric  between  them  and  at  a  given  potential 
difference,  and  again  with  air  between  the  conductors 
and  at  the  same  potential  difference.  The  charge  on  the 
conductors  when  air  separates  them  may  also  be  calculated 
mathematically,  as  will  be  shown  later.  The  table  in  Ar- 
ticle 81  gives  the  values  of  the  dielectric  constant  for  some 
of  the  more  commonly  used  dielectrics. 

71.  Electrostatic  Potential. — It  has  already  been  shown 
that  the  work  done  in  an  electric  circuit  is  equal  to  QV, 
where  Q  is  the  quantity  of  electricity  which  passes  between 
the  points  whose  potential  difference  is  V.  Therefore  the 
potential  difference  between  two  points  may  be  defined  as 
the  work  which  would  be  done  in  moving  unit  quantity  of 
electricity  from  one  of  the  points  to  the  other.  Since  work 
is  equal  to  force  times  distance  and  since  electrostatic 
intensity  is  numerically  equal  to  the  force  exerted  on  unit 
quantity  of  electricity  in  a  dielectric  of  air,  it  follows  that 
electrostatic  intensity  is  equal  to  the  potential  difference 
per  unit  distance  measured  in  the  direction  of  the  electro- 
static force.  When  the  work  is  expressed  in  ergs  and  the 
charge  in  electrostatic  units,  the  p.d.  is  expressed  in  what 
are  called  "  electrostatic  units."  The  intensity  of  an 


ELECTROSTATICS  103 

electrostatic  field  may  therefore  be  expressed  either  in  lines 
of  force  per  square  centimeter,  or  in  electrostatic  units  of  p.d. 
per  centimeter  of  path,  and  they  are  numerically  equal  to 
each  other.  It  has  been  determined  experimentally  that 
1  coulomb  is  equal  to  3x10°  electrostatic  units  of  quantity, 
and  that  1  volt  is  equal  to  1  /300  of  the  electrostatic  unit  of 
potential.  In  practice,  electrostatic  intensity  is  generally 
expressed  in  volts  per  inch,  volts  per  mil,  or  volts  per  cen- 
timeter, of  distance  between  the  two  points  under  con- 
sideration. 

72.  Capacity. — When  two  conductors  A  and  B  are  con- 
nected to  a  source  of  e.m.f.  as  in  Fig.  50,  the  amount  of  elec- 
tricity which  flows  out  of  one  into  the  other  (in  other  words 
the  charge  they  receive)  depends  upon  the  surface  area  of 
the  conductors,  upon  the  distance  between  them,  upon  the 
nature  of  the  dielectric  which  separate  them,  and  upon  the 
value  of  the  e.m.f.  to  which  they  are  connected.  When 
the  first  three  conditions  are  fixed,  the  charge  depends 
only  upon  the  e.m.f.  and  varies  in  direct  proportion  with  it. 
In  such  a  case,  the  charge  which  the  circuit  receives  per  volt 
of  e.m.f.  impressed  upon  it  is  known  as  the  capacity  of  the 
circuit.  Care  should  be  taken  to  think  of  capacity  not 
as  the  amount  of  electricity  contained  by  a  circuit,  nor  as 
the  amount  that  it  will  stand  without  breaking  down, 
but  simply  as  the  charge  which  each  side  of  the  circuit  will 
have  produced  upon  it  by  1  volt  of  potential  difference. 
The  ratio  of  the  capacity  of  a  circuit  of  given  dimensions 
when  the  dielectric  is  other  than  air  to  its  capacity  when 
the  dielectric  is  air,  is  the  dielectric  constant  of  the  given 
material,  as  defined  in  Article  70.  The  unit  of  capacity  is 
called  a  farad;  a  circuit  has  a  capacity  of  one  farad  when  1 
volt  between  its  terminals  produces  a  charge  of  1  coulomb 
on  each  of  its  sides.  Speaking  broadly,  any  two  conduct- 
ors separated  by  a  dielectric  is  an  electric  condenser,  but 
the  term  condenser  is  more  commonly  restricted  to  mean 
two  parallel  sets  of  plates  or  sheets  of  metal  separated  by  a 
thin  sheet  of  dielectric  material. 


1O4  ELECTRIC  AND  MAGNETIC  CIRCUITS 

73.  Capacity  of  a  Parallel  Plate  Condenser.  —  When  a 
parallel  plate  condenser  is  connected  to  a  source  of  potential, 
one  plate  or  set  of  plates  becomes  charged  positively  and 
the  other  negatively  and  electrostatic  flux  is  said  to  pass 
from  one  to  the  other.  Since  there  are  4?r  lines  of  force  from 
each  electrostatic  unit  of  quantity,  there  will  be  47rX3xl09 
electrostatic  lines  from  each  coulomb  of  charge,  or  12?r  X  10°. 
If  Q  is  the  total  number  of  coulombs  on  each  conductor,  and 
A  is  the  surface  area  of  one  plate  in  square  centimeters, 
then  the  flux  density  from  that  plate  will  be  12  ?r  times 
109Q/A.  If  K  is  the  dielectric  constant  of  the  material 
separating  the  plates,  the  electrostatic  field  intensity  will  be 


This  field  intensity  multiplied  by  the  thickness,  d,  of  the 
dielectric  is  the  total  p.d.  between  the  conductors  in  elec- 
trostatic units.  If  the  p.d.  is  expressed  in  volts  then 


V 

~  =300' 


and  the  capacity  in  farads  is 

c    Q          . 

~v     HFC*  - 

This  equation  holds  only  when  the  distance  d  between  plates 
is  so  small  that  the  flux  is  distributed  uniformly  over  the 
surface,  A,  and  the  flux  passing  from  the  edges  of  the  plates 
can  be  neglected.  If  A  is  in  square  inches  and  d  is  in  mils, 
then  the  equation  becomes 

884.2  KA  (6.45)       225  KA  . 
=  10^(0.001X2.54)  =  ~l^d~  farads' 

The  larad  is  such  a  large  unit  that  the  microfarad,  or 
1/1,000,000  of  a  farad,  is  much  more  commonly  used. 
Expressed  in  microfarads,  the  last  formula  becomes 

.,-'••  (,06) 


ELECTROSTATICS 


105 


74.  Condensers  in  Parallel  and  in  Series. — When  con- 
densers are  connected  in  parallel,  the  p.d.  is  the  same  across 
all  of  them  and  the  total  charge  is  the  sum  of  the  charges 
on  the  several  condensers.  Therefore,  the  equivalent 
capacity  of  several  condensers  in  parallel  is  equal  to  the  sum 
of  the  several  capacities.  When  two  or  more  condensers 
are  connected  in  series,  equal  and  opposite  charges  will  be 
induced  on  each  pair  of  plates  connected  together.  Hence 
if  the  condensers  have  capacities  Ci,  €2,  Cs,  etc.,  the  total 
p.d.  will  be 

Q  ,  Q  ,  Q     {I    .  1    ,  l\n 


Hence  the  equivalent  capacity  is 
0  1 


(108) 


75.  Capacity  of  a  Transmission  Line. — An  important 
property  of  alternating  current  transmission  lines  is  their 
capacity.  The  formula  for  such  capacity  is  developed  as 
follows:  Let  Fig.  53  represent  two  parallel  wires  of  radius  r 


FIG.  53. 

and  distance  D  apart  from  center  to  center.  The  wires 
being  long  in  comparison  with  their  distance  apart,  and  far 
apart  in  comparison  with  their  radius,  the  charge  on  each 
wire  may  be  considered  as  uniformly  distributed  over 


106  ELECTRIC  AND  MAGNETIC  CIRCUITS 

the  surface  of  the  wire.  If  there  are  Q  units  of  charge  per 
centimeter  of  length  of  wire  the  flux  per  unit  of  length  will 
be  47rQ.  The  area  over  which  this  flux  is  distributed  at 
distance  x  from  the  center  of  the  wire  is  2wx.  The  electro- 
static intensity  is  therefore  4:irQ/2irx=2Q/x  at  point  p 
due  to  wire  A.  That  due  to  wire  B  is  similarly  2Q/(D—  x) 
The  total  intensity  at  p  is  therefore 


It  was  shown  in  Art.  71  that  electrostatic  intensity  is  equal 
to  drop  in  potential  per  unit  distance,  or 


Therefore  the  potential  difference  between  A  and  B  is 

;D-r  rD-r 

dv=  I        Hdx.  (Ill) 

Substituting  the  value  of  H  from  equation  (109)  we  have 


D~T 


(113) 

'      -2Qloge(D-x)^    ',         (114) 
V  =2Q  [log,  (D-r)  -2  log,  r+loge  (D  -r)],    (115) 

•  .     (116) 


Since  capacity  is  the  charge  per  unit  potential  difference  we 
get 

•y  =C  =—        /—      e^ec^rostatic  units  per  cm.      (117) 
4  log, 


ELECTROSTATICS  107 

or,  reducing  to  microfarads  per  1000  ft. 

r       30.48  XlO3       If         106         1        0.00368 

,OQI      (D~r\    L300X3X10»J-       (D-r\ 
[4X2.3  log  (—)\  ^(-7-) 


or,  in  microfarads  per  mile 

°- 
D- 


C        °-0194  (H9) 

- 


76.  Charging  Current.  —  Since  the  charge  on  a  condenser 
varies  with  the  p.d.  at  its  terminals,  it  follows  that  when- 
ever the  p.d.  across  a  condenser  is  changing  the  charge  on  the 
condenser  is  changing;   that  is,  electricity  flows  toward  one 
side  of  the  condenser  and  away  from  the  other  side;   but 
electricity  in  motion  is  a  current.     The  rate  at  which  the 
charge  changes,  that  is,  the  value  of  the  current  is  pro- 
portional to  the  rate  at  which  the  p.d.  is  changing. 

Since  i  =  dQ/dt  and  Q  =  CV,  we  get  the  value  of  the  cur- 
rent which  flows  into  and  out  of  a  condenser,  due  to  a 
changing  potential  difference,  as 

.  .    (120) 

This  current  is  called  the  displacement  current,  or  the  charg- 
ing current,  to  distinguish  it  from  the  steady  current  which 
we  have  in  a  closed  electric  circuit.  From  this  relation, 
capacity  is  sometimes  defined  as  the  ratio  of  the  displace- 
ment current  of  a  condenser  to  the  rate  of  change  of  p.d. 
which  produces  it. 

77.  Energy  of  a  Condenser.  —  It  follows  from  the  fact. 
that  a  current  is  produced  when  a  p.d.  is  connected  to  a 
condenser,  that  energy  is  being  transformed.     The  amount 
of  energy  which  is  stored  up  in  a  condenser  when  it  is 
charged  is  %CV2.     Multiplying  the  equation  i  =  CdV/dt, 
by  V,  we  get 

Vi  =  CVdV/dt,  (121) 

or 

(122) 


108  ELECTRIC  AND  MAGNETIC  CIRCUITS 

but  Vidt  =  dW,  which  is  the  energy  delivered  to  the  con- 
denser in  the  time  dt,  during  which  the  p.d.  changes  by  an 
amount  dV.  Therefore,  the  energy  delivered  when  the  p.d. 
is  raised  from  0  to  V  is 

W=fQVCVdV=iCV2.  (123) 

78.  Distribution    of    Electrostatic    Intensity. — Electro- 
static lines  of  force  are  to  be  thought  of  as  issuing  from  the 
charge  on  one  conductor  and  passing  to  that  on  the  opposite 
conductor.     The  density  of  these  lines  as  they  issue  from  a 
conductor  will  therefore  depend  upon  the  manner  in  which 
the  electricity  is  distributed  over  the  surface  of  the  con- 
ductor.    In   the    case   of   sheets   of    conducting   material 
separated  by  sheets  of  insulating  material,  all  of  uniform 
thickness,  the  distribution  will  be  uniform  except  at  the 
edges.     It  has  been  found  experimentally  that,  hi  distrib- 
uting itself  over  a  surface,  electricity  always  piles  up,  so 
to  speak,    at   edges,   bends,    and   sharp   points   generally. 
Therefore  the  electrostatic  intensity  is  always  higher   at 
such  points  than  over  the  smooth  portions  of  the   surface 
separating   the   conductor   from   the   dielectric.     Immedi- 
ately after  leaving  such  points,  however,  the  lines,  spread 
out,  or  the  intensity  becomes  less,  and  the  general  distribu- 
tion becomes  more  nearly  uniform. 

79.  Potential  Gradient. — Since  electrostatic  intensity  is 
proportional  to  the  drop  (or  rise)  in  electrostatic  potential 
per  unit  distance,  the  intensity  at  any  point  is  very  com- 
monly expressed  as  the  potential  gradient  at  that  point. 
The  term  potential  gradient  means  the  drop  (or  rise)  of 
potential  per  unit  distance.     In  uniform  fields,  it  is  a  con- 
stant;   in  non-uniform  fields,  it  is  the  derivative,  dv/dx. 
Curves  may  be  plotted  with  potential  difference  as  ordinates 
and  distances  away  from  the  conductor  as  abscissae,  or  they 
may  be  plotted  with  potential  gradient  (dv/dx)  as  ordinates 
and  distances  as  abscissas.     The  potential  difference  referred 
to  here  is  the  potential  difference  between  one  conductor 
and  different  points  along  the  shortest  path  between  it  and 


ELECTROSTATICS 


109 


the  other  conductor.  If  this  potential  difference  curve  be 
plotted  for  the  space  between  two  plates  of  a  condenser, 
the  curve  will  start  at  zero  and  rise  in  a  straight  line  to  a 
value  equal  to  the  potential  difference  between  the  plates. 
The  potential  gradient  curve  would  be  a  horizontal  straight 
line.  If  the  potential  difference  curve  be  plotted  for  the 
space  between  two  long  straight  bare  wires  some  distance 
apart  in  air,  the  curve  will  rise  from  zero  at  one  wire,  quite 
steeply  at  first,  then  slope  upward  less  steeply  until  the 
other  wire  is  approached  when  it  will  again  rise  steeply  to  a 
value  equal  to  the  p.d.  between  the  wires.  The  potential 
gradient  curve  would  start  at  a  high  value  and  slope  steeply 
downward  at  first,  then  less  and  less  steeply  to  a  minimum 
midway  between  wires,  then  rise  again  more  and  more 
steeply  as  the  other  wire  is  approached.  These  curves  are 
shown  in  Fig.  54  for  the  case  of  two  parallel  wires. 


V 
dv 


II 


Distance 


D-r 


FIG.  54. — Potential  and  Potential  Gradient  between  two  Wires  of 
Transmission  Line. 

The  equation  for  the  potential  gradient  curve  in  terms 
of  charge  on  the  conductors  is  equation  (109).  It  may  be 
expressed  in  terms  of  the  potential  difference  (V)  between 
the  wires  by  substituting  the  value  of  Q  from  the  equation 
(116)  into  equation  (109).  This  gives 

*    r        v        ..     „     . 


log.( 


D-r" 


.x(D- 


110  ELECTRIC  AND  MAGNETIC  CIRCUITS 

The  equation  for  potential  difference  between  one  wire 
and  various  points  along  the  shortest  path  between  it  and 
the  other  wire  will  be  found  by  integrating  equation  (124). 
The  constant  of  integration  will  be  found  by  applying  the 
condition  that  v  =  0  when  x  =  r.  The  solution  is 


V\      logex-\oge(D-x)l 
"2L1+loge(D-r)-logerJ- 

This  equation  does  not  hold  for  values  of  x  less  than  r  or 
greater  than  (D—  r). 

80.  Losses  in  Dielectrics.  —  Since  no  material  is  a  the- 
oretically perfect  insulator,  there  will  always  be  an  amount 
of  current  flowing  through  the  substance  of  a  dielectric, 
equal  to  the  product  of  the  impressed  voltage,  F,  and  the 
conductance,  g,  of  the  dielectric.     That  is,  the  leakage  cur- 
rent, as  it  is  called,  is 

ia=gV.  (126) 

The  loss  hi  the  dielectric,  due  to  this  current,  is  of  course 
equal  to  Vig  or  gV2. 

When  a  varying  voltage  is  impressed  on  a  dielectric, 
there  is  found  by  experiment  to  be  an  additional  loss,  that 
is,  more  loss  than  can  be  accounted  for  by  the  leakage  cur- 
rent loss.  This  loss  is  called  dielectric  hysteresis  loss  and  is 
probably  caused  by  lack  of  homogeneity  in  the  material 
and  by  the  energy  required  to  reverse  the  stresses  when  the 
voltage  reverses. 

81.  Dielectric   Strength.  —  It   is  found   that   when   the 
intensity  of  the  electrostatic  field  exceeds  certain  values 
(different  for  different  dielectrics)  the  substance  "  breaks 
down  "  and  allows  the  electricity  to  flow  through  it.     The 
value  of  the  volts  per  unit  thickness  at  which  a  dielectric 
fails  is  known  as  its  dielectric  strength.     In  cases  where  the 
electrostatic   intensity   is   not   uniform,    as,    for   example, 
between    the  wires  of  a  transmission    line,  the    dielectric 
strength  of  the  air  may  be  exceeded  at  points  near  the 
wires  and  not  be  exceeded  in  the  rest  of  the  space  between 


ELECTROSTATICS 


111 


wires.  In  such  cases  the  air  in  the  region  where  the  dielec- 
tric strength  is  exceeded  becomes  a  fairly  good  conductor. 
This  change  in  its  nature  is  accompanied  by  the  appearance 
of  a  bluish  light  along  the  wire.  This  effect  is  called  the 
"  corona  effect  "  or  "  brush  discharge. " 

There  are  several  factors  which  affect  the  dielectric 
strength  of  materials.  In  general  a  thick  piece  of  insulation 
will  break  down  at  a  lower  pressure  per  unit  thickness  than 
will  a  thin  one.  A  piece  built  up  of  several  thin  ones  will 
generally  stand  more  than  an  equally  thick  solid  piece. 
The  chief  reason  is  that  a  solid  thick  piece  is  not  likely  to 
be  so  homogeneous  nor  so  free  from  flaws  as  a  thin  one. 
An  increase  in  temperature  generally  results  in  a  decreased 
dielectric  strength.  A  high  frequency  causes  more  loss  and 
a  higher  temperature  and  consequently  a  lower  strength 
than  a  low  frequency.  When  time  is  taken  in  seconds  or 
perhaps  in  minutes,  a  material  will  stand  a  high  voltage  for 
a  short  time  but  will  break  down  if  the  same  voltage  is 
kept  on  for  a  longer  time.  Other  conditions,  such  as  the 
form  of  the  conductors  and  the  wave  form  of  the  voltage, 
also  affect  the  breakdown  value. 

It  should  be  noted  that  there  is  no  definite  relation  be- 
tween the  dielectric  strength  of  a  material  and  its  resistance 
as  insulation.  The  following  table  gives  characteristic 
values  of  the  properties  of  dielectric  and  insulating  materials : 

TABLE   2. — CONSTANTS   OF   DIELECTRIC   MATERIALS 


Material. 

Dielectric 
Constant. 

Dielectric 
Strength, 
Volts  per  Mil. 

Resistivity, 
Ohm-cm. 

Air 

1 

97 

Glass        

5  to  10 

150  to  300 

17X109 

Mica                                     .  . 

2  5  to6 

1000  to  3000 

(1  to  120)  X1012 

Paraffin 

1  9  to  2  3 

300 

1016  to  1019 

Paper  (dry)  untreated  
Petroleum 

1.7to2.6 
2  to  3 

100  to  250 
200  to  400 

10  15 

1Q12 

Rubber  (para) 

2  to  3 

300  to  500 

1014  to  1016 

112  ELECTRIC  AND  MAGNETIC  CIRCUITS 

82.  Charging  and  Discharging  a  Condenser  through  a 
Resistance. — Attention  has  already  been  called  to  the  fact 
that  when  a  source  of  e.m.f.  is  first  connected  to  two  con- 
ductors separated  by  a  dielectric  (i.e.,  a  condenser)  a  cur- 
rent will  begin  to  flow  but  will  cease  as  soon  as  the  p.d. 
across  the  condenser  becomes  equal  to  the  e.m.f.  which 
has  been  connected  to  it.  The  time  required  to  reach  this 
condition  depends  upon  the  resistance  in  series  with  the 
circuit.  When  the  source  of  e.m.f.  is  removed  and  con- 
denser circuit  is  closed,  the  condenser  will  discharge;  that 
is,  a  current  will  begin  to  flow  but  will  cease  as  soon  as  the 
p.d.  across  the  condenser  becomes  zero,  provided  there  is  no 
inductance  in  the  circuit.  (The  case  of  capacity  and 
inductance  together  will  be  discussed  in  the  next  article.) 
Again,  the  time  required  for  the  current  to  become  zero 
depends  upon  the  value  of  the  resistance  in  series  with  the 
condenser.  Following  is  the  development  of  the  mathe- 
matical relations  between  current  and  time  for  the  two  sim- 
ple cases  just  mentioned. 

Referring  to  Fig.  55,  it  will  be  seen  that  when  the  key  K 

is  in  contact  with  b  the  condenser  and  resistance  are  in 

c  r  series  with  the  source  of  e.m.f. 

E.  With  this  connection,  the 
drop  in  potential,  v,  through 
the  condenser,  plus  the  drop 
in  potential,  n,  through  the 


^zn^L 


|     |£        rr 

\E  resistance  must   be   equal  at 

FlG  55  every  instant  to  the  e.m.f.  E. 

That  this  is  true  should  be 

understood  from  the  general  principle  that  in  any  circuit 
the  total  e.m.f.  in  the  circuit  must  be  equal  to  the  sum  of  the 
potential  drops  which  consume  this  e.m.f.  We  have  then 
as  the  e.m.f.  equation  for  the  circuit 

E=v+ri  (127) 

But  v  =q/C  and  i  =  dq/dt,  where  q  is  the  charge  on  the  con- 
denser at  voltage  v,  and  dq/dt  is  the  corresponding  rate  of 


ELECTROSTATICS  113 

charge  of  q.     Therefore, 


Separating  tne  variables  in  this  equation,  we  get 

rc=cw^q-  <129> 

Integrating  this  equation 

:,  (130) 


when  K  is  the  constant  of  integration.  For  the  case  under 
consideration,  it  is  known  that  q=0  when  t=0,  if  the  zero 
of  time  is  taken  as  the  instant  when  the  connection  is  made 
to  E.  Therefore,  substituting  these  values  in  equation 
(130),  we  get 

K=\ogeCE,  (131) 

and  changing  signs  in  equation  (130)  and  substituting  the 
value  of  K 


-«)  -lo&  CE  =lo&          '   (132) 


or 

'  (133) 


where  e  is  the  base  of  the  natural  system  of  logarithms; 
whence 

ff-O0(l-e  *?).  (134) 

From  this  the  value  of  i  is  at  once  found  by  differentiation 
with  respect  to  t,  as 


114 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


The  curve  for  this  current  is  shown  in  Fig.  56.     Of  course, 
since  no  circuit  can  be  completely  without  inductance,  the 

current  cannot  rise  instantly 
from  zero  to  the  initial  value 
here  shown  as  E/r,  but  the 
effect  of  inductance  has  been 
ignored  in  this  discussion. 

Now  suppose  after  the  con- 
denser is  charged  to  the  poten- 
tial difference,  E,  and  contains 
a  charge  Q,  the  key  in  Fig.  56 
is  allowed  to  make  contact  at 
a.  The  condenser  will  begin 
to  discharge  and  since  the 
e.m.f.  E  has  been  removed, 
the  equation  for  e.m.f  s  becomes 


(136) 


Again  separating  the  variables  and  integrating,  we  get, 


(137) 


In  this  case,  the  constant  of  integration  is  found  from  the 
condition  that  q=Q  at  t=0,  if  the  zero  of  time  is  taken  as 
the  instant  when  connection  is  made  to  a.  We  get,  there- 
fore, K=logQ,  and 


or 
whence 

or,  since 


i-dt-'rC6 
Q  =  CE, 

E     ' 

i=  —e  rC. 


(139) 
(140) 

(141) 


ELECTROSTATICS  115 

The  shape  of  the  current  curve  is  therefore  the  same  as  for 
the  case  of  charging  but  it  flows  in  the  opposite  direction, 
as  is  indicated  by  the  minus  sign. 

83.  Short-circuiting  Inductance  and  Capacity  in  Series. — 
A  mathematical  study  of  transient  phenomena,  that*  is, 
those  phenomena  which  occur  when  there  is  a  change  from  a 
steady  condition,  is  beyond  the  scope  of  this  text.  Two 
exceptions  have  been  made  to  this;  namely,  the  cases  of 
building  up  a  magnetic  field  in  a  simple  inductive  resistance, 
and  of  charging  and  discharging  a  condenser  through  a 
resistance.  One  additional  case  will  be  discussed  briefly; 
that  is  the  case  of  closing  a  circuit  containing  an  inductance 
and  a  condenser,  when  there  is  either  a  current  flowing  hi 
the  inductance  or  a  p.d.  across  the  condenser.  Such  a  cir- 
cuit would  be  represented  by  a  transmission  line,  if  its  capac- 
ity be  considered  as  concentrated  at  some  one  point  and  a 
short  circuit  occurred  at  some  other  point,  opening  the  circuit 
breakers  at  the  power  station.  The  effect  of  resistance  in  the 
circuit  will  at  first  be  neglected.  Assume  that  the  short- 
circuit  occurs  at  an  instant  when  the  line  is  charged  to  a 
potential  difference  of  V  volts  and  the  current  in  the  line  is 
zero.  The  electrostatic  energy  of  the  line  will  be  |CF2 
where  C  is  its  capacity.  Current  will  begin  to  flow  from  one 
side  of  the  line  to  the  other  through  the  short  circuit,  and 
by  building  up  a  magnetic  field  linking  the  circuit,  the  elec- 
trostatic energy  will  be  transformed  into  electromagnetic 
energy.  Since  it  is  assumed  that  none  of  the  energy  is 
dissipated  in  heat  losses,  it  follows  that  when  the  elec- 
trostatic energy  has  become  zero,  an  amount  of  energy  equal 
to  the  original  value  of  the  electrostatic  energy  must  now 
be  stored  in  the  circuit  in  the  form  of  electromagnetic  energy. 
The  energy  stored  in  a  magnetic  field  has  been  shown  to  be 
equal  to  ILP  where  L  is  the  inductance  of  the  circuit.  It 
follows  therefore  that  the  current  in  the  circuit  will  reach 
such  a  value  that  iLP=±CV2,  or  I  =  V^C~/~L.  This  is 
not  likely  to  result  in  a  dangerous  condition;  but  if  the 
short  circuit  occurs  when  the  current  in  the  line  is  I  and 


116  ELECTRIC  AND  MAGNETIC  CIRCUITS 

the  p.d.  across  the  line  is  zero,  then  the  electromagnetic 
energy  associated  with  the  line  will  be  equal  to  f  L/2,  and 
as  the  current  decreases,  this  energy  will  be  transformed 
into  electrostatic  energy,  and  the  p.d.  across  the  line  at  the 
point  where  the  capacity  is  located  will  rise  to  such  a  value 
that  ±CV2=±LP,  of  y  =  /VZ7C.  This  may  result  in  a 
dangerously  high  voltage,  if  the  current  and  inductance  are 
large  and  the  capacity  is  small.  In  either  one  of  the  cases 
mentioned  the  energy  associated  with  circuit  at  the  time  of 
short-circuit  will  continue  to  oscillate  back  and  forth  from 
one  form  to  the  other  indefinitely.  In  any  practical  case, 
there  will  be  resistance  in  the  circuit  which  will  consume  a 
portion  of  the  energy  during  each  oscillation  and  thus  the 
amplitude  of  the  oscillations  will  gradually  decrease  until 
all  of  the  energy  has  been  dissipated  in  heat. 


CHAPTER  VI 
SINE  WAVE  ALTERNATING  CURRENTS 

84.  Definition  of  Alternating  Current.  —  An  alternating 
current    (or  e.m.f.)    is   one   which   alternates   regularly   in 
direction  between  the  same  positive  and  negative  maximum 
values  and  whose  average  value  is  zero  when  taken  over  any 
whole  number  of  cycles  of  values.     In  ordinary  electric 
machinery,  the  positive  sets  of  values  are  exactly  like  the 
negative    sets   of   values.     Furthermore,    these    values,    if 
plotted  against  time  as  abscissae,  give  a  curve  which  approx- 
imates more  or  less  closely  to  what  is  called  a  sine  wave, 
which  is  a  curve  plotted  with  angles  as  abscissae  and  the 
corresponding  sines  of  those  angles  as  ordinates.     In  this 
chapter,  the  entire  discussion  will  be  based  on  the  assump- 
tion that  the  curves  have  the  form  of  sine  waves 

85.  The  E.M.F.  and  Current  Equations.  Cycle.    Fre- 
quency.   Angular  Velocity.    Electrical  Degrees.    Phase.  — 
A  sine  wave  will  be  generated  if  a  coil  of  wire  is  revolved  at 
uniform  speed  in  a  uniform  magnetic  field  as  illustrated  in 
Fig.  57.     The  fundamental  e.m.f.  equation  has  already  been 
shown  to  be 


The  negative  sign  is  placed  before  the  right-hand  member 
because  the  current  which  this  e.m.f.  produces  in  the  cir- 
cuit will  have  a  force  exerted  upon  it  by  the  field,  which  is 
in  opposition  to  the  force  which  causes  the  coil  to  turn.  If 
<f>m  is  the  maximum  flux  which  can  be  enclosed  by  the  coil, 

117 


118 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


then  the  flux  enclosed  in  any  given  position  of  the  coil,  such 

as  c  —  d,  is 

0  =  <i>m  cos  d,  (143) 

where  6  is  the  angle  between  the  plane  of  the  coil  and  a 
plane  at  right  angles  to  the  direction  of  the  field.  But  if 
the  coil  is  rotating  at  uniform  angular  velocity  of  co  radians 
per  second,  and  time,  t,  be  counted  from  the  instant  of  maxi- 
mum enclosure,  then  8  =  at  and 


=  <f>m  COS 


(143a) 


Position  of  Maximum 
Enclosure  of  Flux 
Cry^aBd  of  Zero  E.M.F. 


Substituting  this  value  of  <£  into  equation  (142)  gives 
Nd 


e=~ 


COS     '  = 


sm 


(144) 


In  equation  (144)  the  part  coN0m/108  is  the  maximum  value 
of  the  e.m.f.  and  is  the  value  at  the  instant  when  ut  is  90° 
or  when  the  flux  linking  the  coil  is  zero;  at  this  instant  the 
coil  is  cutting  across  the  flux  most  rapidly,  whereas,  when 
the  flux  linking  the  coil  is  a  maximum,  the  coil  is  sliding 
along  the  flux  and  the  rate  of  cutting  is  zero.  Denoting 
the  maximum  e.m.f.  by  Em,  we  may  write 

e  =  Em  sin  ut.  (145a) 

Similarly  we  may  write  the  equation  for  the  instantaneous 
value  of  current  in  a  circuit  as 


7™  sin 


(1456) 


SINE  WAVE  ALTERNATING  CURRENTS         119 

where  t  =0,  when  i=0.  The  successive  values  ol  the  e.m.f. 
for  one  revolution  of  the  coil,  beginning  at  t  =  0,  would  be  as 
shown  in  Fig.  58.  One  complete  set  of  values  is  called  a 
cycle.  One  cycle  of  values  is  passed  through  for  each  revo- 
lution of  the  coil  in  a  2-pole  field  such  as  is  shown  in  Fig.  57. 
During  the  first  half  revolution  the  e.m.f.  is  in  one  direction 
in  the  wire  and  is  indicated  by  ordinates  above  the  o>axis 
in  Fig.  58,  while  during  the  second  half  of  the  revolution,  the 
e.m.f.  is  in  the  opposite  direction  in  the  wire  and  is  indicated 
by  ordinates  below  the  x-axis.  Either  direction  through  the 
wire  may  be  chosen  as  positive.  The  curve  showing  the 


FIG.  58. 

values  of  e.m.f.  or  current  as  a  function  of  time  is  called 
the  wave  form.  The  time  consumed  in  passing  through 
one  cycle  is  called  the  period,  and  the  number  of  cycles 
passed  through  in  one  second  is  called  the  frequency;  in 
a  2-pole  field  the  frequency  (/)  is  equal  to  the  number  of 
revolutions  per  second,  and  the  angular  velocity  of  the 
coil  is  equal  to  2ir  times  the  frequency,  or 

0)=27T/.  (146) 

In  a  4-pole  field  such  as  is  illustrated  in  Fig.  59,  there 
will  be  two  cycles  of  e.m.f.  for  one  revolution  of  the  coil, 
and  the  frequency  will  be  equal  to  the  number  of  pairs  of 
poles  times  the  number  of  revolutions  per  second,  or,  in 


120 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


general,  if  p  is  the  number  of  poles  and  n  is  the  number  of 
revolutions  per  minute,  the  relation  is 


/= 


pn        pn 
2^X60  =I20' 


(147) 


Occasionally  the  term  "  alternations "  is  used  in  con- 
nection with  an  alternating  current.  This  refers  to  the 
number  of  reversals  per  minute  and  is  therefore  equal  to 
120  times  the  frequency  or  to  the  number  of  poles  times  the 
revolutions  per  minute. 


FIG.  59. 

In  equation  (144)  the  quantity  ut  is  an  angle  and  the 
e.m.f.  passes  through  one  cycle  while  the  angle  changes 
from  0  to  360°,  or  from  0  to  2?r  radians;  and  one  cycle  is 
always  passed  through  for  every  pair  of  poles  which  is  passed; 
therefore  in  electrical  machinery  the  angle  covered  by  each 
pair  of  poles  is  2?r  (electrical)  radians  or  360  (electrical) 
degrees.  The  number  of  electrical  degrees  passed  over  in 
one  revolution  is  p/2  times  the  mechanical  degrees  passed 
over.  The  angle  swept  through  is  a  function  of  time.  In 
the  ground  covered  by  this  text,  steady  conditions  of  opera- 
tion are  assumed  and  therefore  the  angular  velocity  of  the 
moving  elements  of  generating  apparatus  is  constant,  and 


SINE  WAVE  ALTERNATING  CURRENTS         121 

the  angle  swept  over  in  any  given  time  is  equal  to  the  product 
of  that  time  and  the  angular  velocity.  Furthermore,  under 
the  conditions  assumed,  the  recurring  cycles  of  values  are 
exactly  alike.  It  is  therefore  permissible  and  convenient 
to  express  the  instantaneous  values  of  e.m.f .  and  current  as  a 
function  of  an  angle,  as  has  been  done  in  the  preceding 
discussion. 

It  will  be  proven  in  later  articles  that  in  circuits  contain- 
ing inductance  or  capacity  the  current  and  e.m.f.  will  not 
have  their  maximum  values  (nor  any  other  corresponding 
values)  occurring  at  the  same  instant.  This  condition  is 
described  by  saying  that  the  current  and  e.m.f.  are  "  out  of 
phase."  If  the  e.m.f.  equation  is  written  as  e  =  Em  sin  ut, 
then  the  instantaneous  value  of  e  is  zero  when  t=0j  and  if 
the  current  is  out  of  phase  with  the  e.m.f.,  the  current  will 
not  be  zero  when  t  =0.  Consequently,  in  order  to  give  cor- 
rect corresponding  values,  the  current  equation  must  be 
written  as 

i  =  /msin  (orf-0)  (148) 

in  which  Im  is  the  maximum  value  of  the  current  and  <f>  is 
an  angle  whose  value  is  u(tf  —  t),  where  (t' —  t)  is  the  con- 
stant difference  in  time  between  corresponding  values  of  e 
and  i.  The  angle  0  is  called  the  angle  of  phase  difference 
and  in  practice  the  phase  difference  is  expressed  in  terms  of 
an  angle  instead  of  time.  Note  that  when  <t>  is  a  positive 
angle,  the  e.m.f.  passes  through  zero  in  a  positive  direction 
before  the  current,  or,  as  it  is  generally  expressed  the  cur- 
rent "  lags  "  the  e.m.f.,  see  Fig.  58.  For  instance,  when 
t=0,  e=0,  and  i  =  Im  sin  (  —  </>);  that  is,  i  is  still  negative 
and  will  not  become  zero  until  o>£  =  $.  On  the  other  hand, 
if  </>  is  negative,  then  when  t=0,  i  =  Imsm  0,  and  has  passed 
through  its  zero  value  before  the  e.m.f.  In  such  a  case,  the 
current  is  said  to  "  lead  "  the  e.m.f.  If  in  the  expression 
(t'-t\  which  is  defined  above,  we  put  t  =  0,  then  (tr —  t)  =t' 
and  t'  is  the  time  that  elapses  between  a  zero  value  of  e  and 
the  nearest  zero  value  of  i  in  the  same  direction  and  it  may 


122  ELECTRIC  AND  MAGNETIC  CIRCUITS 

be  either  positive  or  negative,  depending  on  whether  the 
current  has  not  yet  reached  its  zero  value  or  has  already 
passed  it. 

It  should  be  noted  that  two  or  more  e.m.f.s  or  two  or 
more  currents  may  be  out  of  phase  with  each  other,  and  their 
phase  differences  would  be  shown  in  the  same  manner  as 
the  phase  difference  between  an  e.m.f.  and  a  current. 

86.  Effective  and  Average  Values  of  Current  and  E.M.F. 
—The  power  which  is  developed  when  a  current  flows  in  a 
resistance  R  is  equal  at  each  instant  to  i2R,  or  e2/R,  where 
e  =  Ri.  The  average  power  is  the  average  of  the  instan- 
taneous values.  The  constant  value  of  current  or  e.m.f. 
which  would  produce  an  amount  of  power  in  a  given  resist- 
ance equal  to  the  average  power  produced  by  the  alternating 
wave  is  called  the  effective  value  of  the  alternating  current 
or  e.m.f.  That  is,  if  the  effective  value  is  represented 
by  7,  the  product  RI2  is  equal  to  the  average  power  devel- 
oped by  the  alternating  current,  or, 

RI2  =  average  (Ri2)  =  average  (RI 2  sin2  «J)       (149) 
or 

72  =  I2  (average  sin2  «0-  (150) 

1     cos  2  ut 
But  sin2  wt  =  ~  —  — -o — ,  and  the  average  value  of  cos  2  co£ 

£          £ 

over  any  whole  number  of  cycles  is  zero.  Therefore,  the 
average  value  of  sin2  wt  over  any  whole  number  of  cycles 
is  1/2,  and 

/2=f,        '  (151) 

or 

..Hi-7077*       •'  (152) 

That  is,  the  effective  value  of  an  alternating  current  is 
equal  to  its  maximum  value  divided  by  V2. 

Similarly,  the  effective  value  of  an  alternating  e.m.f.  is 

(153) 


SINE  WAVE  ALTERNATING  CURRENTS          123 

In  practical  work  effective  values  are  nearly  always  of  most 
importance  and  all  measuring  instruments  are  graduated 
to  read  effective  values. 

The  average  value  of  an  alternating  current  is  taken  as 
the  average  ordinate  of  any  half-cycle,  since  the  average 
value  over  a  whole  cycle  is  evidently  zero.  Since  the  instan- 
taneous value  is 

i  =  Im  sin  ut, 
Iav  =  Im\  average  value  of  sin  (wt)\  .  (154) 

But  the  average  value  of  the  sine  of  an  angle  varying  between 
the  limits  of  0  and  TT  is 

1  C*  1  I71"     2 

average  sine  =  -  I     sin  (u()d(<a£)  =  —-  cos  (at)     =-.      (155) 

TTjQ  7T  JO          7T 

Therefore 

(156) 


The  ratio  of  the  effective  value  to  the  average  value  of  an 
alternating  current  or  e.m.f.  is  called  form  factor.  This 
ratio  is,  for  sine  waves,  .707/.637  =  1.11.  The  ratio  of  the 
maximum  to  the  effective  value  is  called  the  crest  factor,  or 
peak  factor,  and  for  sine  waves  it  is  V2  =  1.414. 

87.  Current  and  E.M.F.  Waves  in  Resistance  Only.— 
If  the  instantaneous  current  in  a  resistance  R  is  i  =  Im 
sin  wtj  the  e.m.f.  at  each  instant  must  be  equal  to  Ri  and 
therefore  the  e.m.f.  wave  is  represented  by  the  equation, 

e  =  Ri  =  Rim  sin  ut  =  Em  sin  wt.  (157) 

The  maximum  e.m.f.  is  equal  to  RIm  and  the  effective  value 
of  e.m.f.  is 

E  =  RI  (158) 

or  the  effective  current  is 

I -  (159) 


124 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


The  current  is  said  to  be  "  in  phase  "  with  the  e.m.f.; 
that  is,  they  pass  through  zero  in  the  same  direction  at  the 
same  instant.  These  relations  are  shown  in  Fig.  60. 

88.  Current  and  E.M.F.  Waves  in  Inductance  Only.— 
It  has  previously  been  shown  that  the  e.m.f.  produced  by  a 
varying  current  in  an  inductive  circuit  is  —Ldi/dt.  The 
negative  sign  indicates  the  fact  that  the  e.m.f.  of  self- 
induction  opposes  the  change  in  the  value  of  the  current. 
Hence,  to  maintain  a  current  in  an  inductive  circuit,  there 
will  be  required  an  impressed  e.m.f.  equal  and  opposite  to 
the  e.m.f.  of  self-induction,  and  its  equation  will  be 


di 


(160) 


FIG.  60. 


Therefore,  if  the  equation  for  the  current  is  i  =  Im  sin  at 
the  equation  for  the  e.m.f.  will  be 


d(Im  sin 


e  = 


cos 


or 


msin  M+90) 


(161) 
(162) 
(163) 


The  interpretation  of  this  equation  is  that  the  e.m.f. 
leads  the  current  by  90°,  or  a  quarter  of  a  cycle;  that  is,  the 
e.m.f.  is  a  positive  maximum  when  the  current  is  passing 
through  zero  in  the  positive  direction.  •  This  is  also  ex- 
pressed by  saying  that  the  current  is  out  of  phase  with  the 
e.m.f.  by  90°,  or  that  the  current  lags  the  e.m.f.  by  90°. 
More  strictly,  difference  in  phase  means  difference  in  time 


SINE  WAVE  ALTERNATING  CURRENTS          125 

between  successive  corresponding  values  of  the  alternating 
quantities;  but  as  already  explained,  time  in  connection 
with  alternating  quantities  is  usually  expressed  in  terms  of 
an  angle,  and  therefore  phase  difference  is  usually  expressed 
in  degrees.  In  the  case  under  consideration,  this  phase  dif- 
ference of  90°  is  caused  by  the  back  e.m.f .  of  self-induction. 


FIG.  61. 

One-half  the  tune,  that  is,  while  the  current  is  rising,  the 
current  is  in  the  same  direction  as  the  impressed  e.m.f.; 
during  the  other  half  of  the  time,  that  is,  while  the  current 
is  decreasing,  it  flows  in  the  opposite  direction  to  that  of  the 
impressed  e.m.f.  See  Fig.  61. 

A  clearer  understanding  of  this  matter  may  be  had  by 
studying  an  analogy.     In  Fig.  62  let  it  be  considered  that 


PI 


FIG.  62. — Analogy  of  Inductive  Circuit. 

the  cylinder  C  and  the  pipe  mm  are  full  of  water  and  that  the 
water  is  made  to  flow  back  and  forth  through  the  circuit 
by  the  piston  P  which  moves  in  simple  harmonic  motion  if 
the  drive  wheel  W  revolves  at  uniform  speed.  Since  the 
electric  circuit  which  is  being  discussed  is  assumed  to  be 
without  resistance,  the  frictional  resistance  to  the  flow  of 


126  ELECTRIC  AND  MAGNETIC  CIRCUITS 

water  can  be  neglected  in  the  analogy.  The  inductance  of 
the  electric  circuit  is  analogous  to  the  inertia  of  the  water 
in  the  water  circuit.  When  the  piston  is  in  the  position 
Po,  its  speed  and  also  that  of  the  water  is  a  maximum  and 
at  the  instant  is  neither  increasing  nor  decreasing.  There- 
fore no  pressure  (corresponding  to  voltage  in  the  electric 
circuit)  will  be  exerted  on  the  water,  although  the  water 
flow  (corresponding  to  the  electric  current)  is  a  maximum. 
As  the  piston  continues  to  move,  say  toward  the  position 
PI,  its  speed  decreases  and  the  inertia  of  the  water  begins 
to  exert  a  pressure  upon  it  toward  the  left;  the  pressure  of 
the  piston  upon  the  water  will  therefore  be  from  left  to 
right,  although  the  water  flow  is  in  the  opposite  direction. 
The  pressure  will  become  a  maximum  when  the  piston 
reaches  the  position  PI  because  at  this  position  the  rate 
of  change  of  water  flow  is  greatest  and  its  inertia  is  therefore 
greatest.  The  actual  rate  of  flow  at  this  instant  is  zero, 
however,  and  as  the  piston  begins  to  move  to  the  right,  its 
inertia  will  oppose  the  increase  in  rate  of  flow.  The  pressure 
of  the  piston  will  therefore  be  in  the  same  direction  as  the 
flow  of  water  until  the  piston  again  reaches  the  position  PO, 
when  the  conditions  will  repeat  themselves  as  already  de- 
scribed, while  the  piston  moves  to  P%  and  back  to  P0. 
A  careful  study  of  this  analogy  should  make  clear  what  is 
meant  by  the  statement  that  an  electric  current  is  out  of 
phase  with  its  e.m.f.,  and  how  a  current  may  at  certain 
times  be  flowing  in  the  opposite  direction  to  the  e.m.f. 
impressed  on  the  circuit. 

In  an  inductive  circuit,  resistance  being  ignored,  there 
is  no  opposition  to  the  flow  of  current,  but  there  is  opposi- 
tion to  a  change  in  the  value  of  the  flow.  This  opposition  is 
due  to  the  back  e.m.f.  generated  by  the  accompanying  change 
in  flux  linking  the  circuit;  more  briefly,  it  is  said  to  be  due 
to  inductance.  Referring  to  Fig.  61,  when  the  current  is  a 
maximum  its  rate  of  change  is  zero,  the  rate  of  change  of 
flux  is  also  zero  and  there  is  no  back  e.m.f.  of  self-induction; 
consequently  at  this  instant  no  impressed  e.m.f.  is  required. 


SINE  WAVE  ALTERNATING  CURRENTS          127 

As  the  current  decreases  in  value,  the  flux  also  decreases 
and  a  back  e.m.f.  is  generated  which  opposes  the  decrease 
and  is  consequently  in  the  same  direction  as  the  current". 
Therefore  there  must  be  an  impressed  e.m.f.  equal  and  oppo- 
site to  the  back  e.m.f.  at  each  instant  and  consequently  in 
opposition  to  the  direction  of  the  current.  The  curve  e 
in  Fig.  61  is  that  of  the  impressed  e.m.f.  As  the  current 
decreases  in  value,  its  rate  of  change  increases  and  the 
e.m.f.  increases.  When  the  current  reaches  zero,  it  is  chang- 
ing most  rapidly,  the  back  e.m.f.  is  therefore  a  maximum 
and  the  impressed  e.m.f.  must  be  a  maximum.  As  the 
current  increases  from  zero,  the  back  e.m.f.,  since  it  opposes 
the  rise,  is  opposite  in  direction  and  the  impressed  e.m.f. 
must  be  in  the  same  direction  as  the  current,  but  decreases 
in  value  until  the  current  is  again  a  maximum. 

Returning  to  the  discussion  of  equation  (163),  it  should 
be  noted  that  it  shows  that  the  maximum  e.m.f.  (Em)  is 
equal  to  uLIm  and  it  follows  that  the  effective  e.m.f.  is 

E  =  ^LI=2irfLI.  (164) 

The  product  (2irfL}  is  called  inductive  reactance  and  is 
measured  in  ohms  just  the  same  as  resistance. 

The  value  of  the  current  7  which  will  flow  in  a  purely 
inductive  circuit  of  inductance  L,  when  an  e.m.f.,  E,  of 
frequency  /  is  impressed  upon  it  is,  therefore 

(165) 


and  it  lags  90°  behind  the  e.m.f. 

89.  Current  and  E.M.F.  Waves  in  Capacity  Only.—  It 
has  previously  been  shown  that  the  current  which  flows 
through  a  condenser  when  the  e.m.f.  across  it  is  varying  is 
i=dQ/dt  =  Cde/dt;  this  is  the  instantaneous  value  of  the 
current  ;  hence  to  maintain  a  sine  wave  of  current  through  the 
condenser,  the  relation  must  be 

i-C./.fliatrf,  (166) 


128 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


or 


de  =  -£  sin  utdt. 
Integrating  both  sides  of  this  equation  we  get, 


or 


e=  — ^  cos  ut, 


=  ~  sin  M-90), 


(167) 

(168) 
(169) 


This  equation  shows  that  the  current  leads  the  e.m.f.  by 
90°,  or  a  quarter  of  a  cycle;  that  is,  the  e.m.f.  is  a  negative 
maximum  when  the  current  is  passing  through  zero  in  the 
positive  direction.  See  Fig.  63.  It  also  shows  that  the 


FIG.  63. 

maximum  e.m.f.  is  Im/wC  and  it  follows  that  the  effective 
e.m.f.  is 

E  =  I/uC  =  I/2itfC.  170) 

The  quantity,  l/2ir/C  is  called  capacity  reactance  and  is 
measured  in  ohms.  The  effective  current  hi  a  condenser 
is  therefore 

7  =  27r/C#,  (171) 

and  it  leads  the  e.m.f.  by  90°.  Equations  (169)  and  (163) 
should  be  compared  and  also  equations  (171)  and  (165) 
and  the  differences  carefully  noted  and  understood. 

An  analogy  which  may  make  the  relations  in  a  condenser 
(capacity)  circuit  clearer  is  illustrated  in  Fig.  64.  Suppose 
that  the  cylinder  C,  the  pipes  mm,  and  the  chamber  B  are 


SINE  WAVE  ALTERNATING  CURRENTS 


129 


full  of  water  and  that  the  water  is  made  to  flow  back  and 
forth  through  the  circuit  by  the  simple  harmonic  motion  of 
the  piston  P,  when  the  wheel  W  revolves  at  uniform  speed. 
Let  it  be  imagined  that  the  water  has  no  inertia  and  that  there 
is  no  frictional  resistance  to  its  motion.  Let  D  be  an  imper- 
vious elastic  diaphragm  stretched  across  the  chamber  B. 
The  elasticity  of  this  diaphragm  is  analogous  to  the  capacity 
of  a  condenser.  When  the  piston  is  in  the  position  P0,  its 
speed  and  also  that  of  the  water  is  a  maximum  and  at  the 
instant  is  neither  increasing  nor  decreasing.  Therefore  no 
pressure  (corresponding  to  the  voltage  on  a  condenser)  will 
be  exerted  on  the  diaphragm,  although  the  flow  of  water 


FIG.  64. — Analogy  of  Capacity  Circuit. 

(corresponding  to  the  electric  current)  is  a  maximum.  As 
the  piston  continues  to  move,  say  toward  the  position 
PI,  its  speed,  and  with  it  the  rate  of  flow  of  water,  decreases, 
but  an  increasing  pressure  will  be  exerted  on  the  diaphragm 
in  the  direction  of  the  flow  of  water,  and  this  pressure  will 
reach  its  maximum  when  the  piston  reaches  the  end  of  its 
stroke  and  the  rate  of  water  flow  (current)  is  zero.  As  the 
piston  moves  back  toward  the  position  PQ  the  direction  of 
water  flow  will  be  reversed  and  increase,  but  the  pressure 
on  the  diaphragm  will  continue  as  before  until  the  middle 
position  is  reached,  when  it  will  become  zero  and  then  in- 
crease in  the  same  direction  as  the  water  flow.  A  study  of 
these  relations  should  make  it  clear  that  the  rate  of  water 
flow  in  a  given  direction  is  one-quarter  of  a  cycle  ahead  of 


130  ELECTRIC  AND  MAGNETIC  CIRCUITS 

the  pressure  in  the  same  direction.  .This  is  analogous  to 
the  condition  that  exists  when  an  alternating  voltage  is 
impressed  on  an  electric  condenser. 

Referring  to  Fig.  63,  when  the  e.m.f.  impressed  on  the 
condenser  is  zero,  it  is  changing  most  rapidly  and  therefore 
the  current,  that  is,  rate  of  change  of  charge,  is  a  maximum. 
As  the  p.d.  between  the  plates  of  the  condenser  increases, 
the  flow  of  charge,  that  is,  the  current,  is  in  the  same  direc- 
tion as  the  e.m.f.  across  the  condenser,  but  it  decreases 
in  value  until  the  e.m.f.  reaches  a  maximum  when  the  rate 
of  change  of  e.m.f.,  and  therefore  the  current,  is  zero.  As 
the  e.m.f.  decreases  in  value,  the  charge  flows  out  of  that  side 
toward  which  the  e.m.f.  is  directed  and  is  therefore  negative 
with  respect  to  the  e.m.f.  It  reaches  its  negative  maximum 
when  the  e.m.f.  has  fallen  to  zero,  and  as  the  e.m.f.  rises  in 
the  negative  direction,  the  current  decreases  in  value  as 
before. 

90.  Vector  Representation  of  Alternating  Quantities.— 
The  great  majority  of  alternating  current  problems  involve 
the  addition  or  subtraction  of  alternating  e.m.f's  or  cur- 
rents. Much  simpler  than  that  of  adding  or  subtracting 
the  plotted  waves  is  the  method  known  as  the  vector  method. 
In  Fig.  65  let  the  waves  (e\)  and  e2)  represent  two  alter- 
nating e.m.f's  differing  in  phase  by  the  angle  6  and  the  sum 
of  which  gives  the  wave  es.  Note  that  wave  (62)  lags 
behind  wave  (ei).  At  the  left,  let  the  lines  OE\  and  OE2 
represent  the  maximum  values  of  waves  (e\)  and  (62), 
respectively,  the  line  OE2  being  placed  0  degrees  behind 
(clockwise  is  behind)  OEi.  Draw  OES  as  the  diagonal  of  a 
parallelogram  on  OEi  and  OE2  as  sides.  These  lines  OEi, 
OE2  and  OES  are  vectors,  because  they  not  only  represent 
the  magnitude  of  the  e.m.f's  but  also  by  their  positions 
represent  the  relative  directions  of  the  e.m.f's.  Let  all 
these  vectors  be  supposed  to  revolve  positively  (counter- 
clockwise) at  an  angular  velocity  of  w.  Then  the  ordinates 
of  the  waves  (e\\  (e2),  and  (es)  at  any  point  a  which  is  ut 
degrees  to  the  right  of  the  origin  will  be  respectively, 


SINE  WAVE  ALTERNATING  CURRENTS 


131 


EI  sin  <i>r,  E-2  sin  (o>£  —  0),  and  E8  sin  (ut  —  0),  where  <j>  is  the 
angle  which  OES  makes  with  QE\  and  wZ  is  measured  posi- 
tively (counter-clockwise)  from  the  axis  Ox.  That  is,  OES  is 
the  maximum  value  of  the  resultant  wave,  (es),  and  <f>  is  its 
phase  relation  tc  wave  (e\).  These  two  quantities  (Es  and 
0)  fully  determine  and  locate  the  resultant  wave.  The  value 
of  OES  is  (see  the  triangle  ObEs). 


and 


cos  B 

E2  sin  0 
-' 


sn 


(172) 
(173) 


FIG.  65. 

It  is  therefore  unnecessary  to  plot  and  add  the  waves,  but  the 
resultant  may  be  found  by  the  use  of  the  vector  diagram 
and  its  mathematical  solution.  It  should  be  noted  that 
vectors  cannot  be  used  in  the  manner  just  described  when 
the  quantities  have  different  frequencies,  because  they 
would  then  have  to  revolve  at  different  speeds.  Resultant 
values  and  phase  angles  can  only  be  shown  by  vector  dia- 
grams when  the  vectors  are  stationary  with  respect  to 
each  other;  that  is,  they  must  revolve  at  the  same  speed 
and  therefore  represent  waves  of  the  same  frequency. 

In  the  solution  of  problems  dealing  with  quantities  of  the 
same  frequency  any  set  of  vectors  may  be  considered  as 
actually  as  well  as  relatively  stationary  in  any  desired 
position,  but  the  angles  between  the  different  vectors  must 


132 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


correspond  to  their  actual  phase  relation,  and  it  must  be 
kept  in  mind  that  the  vectors  represent  quantities  which  are 
continuously  changing  in  value  and  alternating  in  direction. 

Since  the  relation  of  the  maximum  values  of  alternating 
quantities  to  their  effective  values  is  a  constant  (V2),  and 
since  effective  values  are  generally  of  most  importance,  the 
latter  values  are  nearly  always  used  in  vector  diagrams. 

Vector  diagrams  are  used  not  only  for  finding  resultant 
e.m.f's  or  currents,  but  also  for  showing  the  phase  relations 
between  e.m.f's  and  currents.  Fig.  66,  shows  the  vector 


FIG.  66. 


E 


FIG.  67. 


FIG.  68. 


diagram  for  the  case  of  Article  87,  Fig.  67  for  the  case  of 
Article  88,  and  Fig.  68  for  the  case  of  Article  89.  These 
diagrams  should  be  carefully  observed  and  the  significance 
of  the  relative  positions  of  the  E  and  7  vectors  for  each  case 
well  understood. 

91.  Current  and  E.M.F.  Relations  in  a  Circuit  Contain- 
ing Resistance,  Inductance  and  Capacity  in  Series. — The 
circuit  will  be  as  shown  in  Fig.  69.  Let  the  current  be 
i  =  Imsm  d>£.  Then  by  equation  (157) 


sn 


by  equation  (162) 


cos 


SINE  WAVE  ALTERNATING  CURRENTS 

and  by  equation  (168) 

Im 

Cc  —  —~7i  COS  (DC. 
wG 

The  total  required  e.m.f.  will  therefore  be 


133 


RIm  sin 


m  cos  wZ  —  —  ?  cos  ut. 


(174) 


It  must  be  remembered  that  this  equation  gives  the 
instantaneous  values  of  the  e.m.f.  The  effective  value 
and  its  phase  relation  to  the  current  can  be  most  readily 
determined  by  means  of  a  vector  diagram.  In  Fig.  70, 
let  the  vector  I  represent  the  effective  value  of  the  current; 


E 


R 


L  C 

FIG.  69. 


7 

WC 


m 


FIG.  70. 


then  the  vector  RI,  drawn  in  phase  with  7,  will  represent 
the  effective  value  of  RIm  sin  <*)£;  the  vector  o)L7,  drawn  90° 
ahead  of  7,  will  represent  the  effective  value  of  wL7m  cos  o>£; 
and  the  vector  7/wC  drawn  90°  behind  7,  will  represent  the 

effective  value  of    ——7^   cos   ut.     From  the  geometry  of 

G)O 

the  diagram,  the  resultant  of  these  three  vectors  is  readily 
found  to  be 


or 


E  = 


L-  — 


(175) 


(176) 


The  terms  wL  and  1/wC  have  already  been  defined  as  "  in- 
ductive reactance  "  and  "  capacity  reactance  "  respectively. 


134  ELECTRIC  AND  MAGNETIC  CIRCUITS 

The  quantity  (wL  —  1/wC)  is  the  total  reactance  and  is 
equivalent  to  a  simple  inductive  reactance  or  a  simple  capac- 
ity reactance  depending  on  whether  wL  is  larger  or  smaller 
numerically  than  1/wC.  Let  the  quantity  (coL  —  1/wC) 
be  represented  by  X.  It  may  be  called  the  "  equivalent 
reactance  "  of  the  circuit.  Equation  (176)  may  then  be 
written 

E  =  IVR*+X2,  (177) 

or 

E 


where  X  may  be  either  positive  or  negative.  The  vector 
diagram  shows  that  E  leads  /  by  an  angle,  <£,  whose  tangent 
is 


-        /RI  =XI/RI, 
or 

0=tan-^.  (179) 

When  inductive  reactance  predominates,  X  is  positive,  </> 
is  positive  and  E  actually  leads  /;  when  capacity  reactance 
predominates,  X  is  negative,  </>  is  negative  and  E  leads  I  by 
a  negative  angle  ;  that  is,  it  lags  behind  7,  or,  what  amounts 
to  the  same  thing,  I  .leads  E.  The  quantity  VR2+X2 
is  called  the  impedance  of  the  circuit  and  is  expressed  in 
ohms  as  are  resistance  and  reactance.  The  letter  Z  is 
universally  used  to  represent  impedance,  and  thus  the 
simplest  expression  for  the  current  in  an  a.  c.  circuit  is 

I  =  |.  (180) 

Evidently,  if  the  circuit  contains  no  capacity,  X  =  wL, 
and  if  the  circuit  contains  no  inductance,  X=  —  l/w(7. 

92.  Effective  Resistance.  —  When  a  direct  current  7 
flows  in  circuit  of  resistance  R}  the  power  dissipated  in  heat 
is  RP.  If  the  current  is  changed  to  an  alternating  one  of 


SINE  WAVE  ALTERNATING  CURRENTS         135 

the  same  effective  value  in  the  same  circuit,  the  power  dis- 
sipated in  heat  will  in  general  be  larger  than  that  which  was 
caused  by  the  direct  current.  This  may  be  due  to  one  or 
more  of  several  causes:  (a)  skin  effect  (see  Article  67); 
(&)  eddy  currents  in  the  conductors;  (c)  hysteresis  in  any 
magnetic  material  associated  with  the  circuit;  or,  (d)  eddy 
currents  in  any  metallic  material  within  the  influence  of  the 
circuit.  These  additional  heat  losses  are  equivalent  in 
their  effect  to  an  increase  in  the  resistance  of  the  circuit. 
The  resistance  to  direct  current  is  called  ohmic  resistance; 
that  value  of  resistance  which  multiplied  by  I2  gives  the 
total  heat  loss  when  alternating  current  flows  in  the  circuit 
is  called  the  effective  resistance  of  the  circuit.  Therefore 
the  effective  resistance  of  an  alternating  current  circuit 
must  be  found  by  measuring  the  total  power  dissipated  as 
heat  and  dividing  this  power  by  the  square  of  the  current. 
In  all  alternating  current  formulae,  the  resistance,  R,  is  to 
be  understood  to  mean  effective  resistance.  In  certain  cases, 
however,  the  hysteresis  and  eddy  current  losses  in  iron  cores 
belonging  to  the  circuit  are  measured  and  considered  sep- 
arately. 

93.  Power  in  A.  C.  Circuits.  —  Let  an  effective  e.m.f. 
E  be  causing  an  effective  current  I  to  flow  through  a  circuit 
of  impedance 


The  angle  of  phase  difference  between  E  and  7  will  be 
4>=tan-l(X/R). 

The  average  power  expended  in  the  circuit  is 

P  =  EIcos<f>.  (181) 

This  may  be  proven  as  follows  :  Let  the  instantaneous  e.m.f. 
be 

e  =  Em  sin  ut  =  V2E  sin  <4  (182) 

Then  the  instantaneous  current  will  be 

i  =  Im  sin  (ut  -  0)  =  V2I  sin  (wZ  -  0),         (183) 


136  ELECTRIC  AND  MAGNETIC  CIRCUITS 

and  the  instantaneous  power  will  be 

p=ei  =  2EI  sin  w£  sin  (ut  -  </>).  (184) 

Substituting  o>2=a,  and  (w£  —  0)=6,  in  the  trigonometrical 
relation  that 

2  sin  a  sin  b  =  cos  (a  —  b)  —cos  (a-f-o),  (185) 

equation  (184)  becomes 

p=EI  cos  <£-#/  cos  (2wZ-0).  (186) 

The  average  power  is  evidently  equal  to  #7  cos  </>  minus  the 
average  value  of  El  cos  (2ut  —  0);  but  the  average  value  of 
any  cosine  or  sine  wave,  over  a  whole  number  of  cycles,  is 
zero.  Therefore  the  average  power  in  the  circuit  is 

P  =  EIco$<t>.  (187) 

In  Figs.  71,  72,  and  73  are  shown  the  voltage,  current 
and  power  waves  for  the  three  cases,  (1)  a  circuit  containing 


FIG.  71. 

R  only,  (2)  a  circuit  containing  R  and  X,  and  (3)  a  circuit 
containing  X  only.  In  all  cases  the  power  wave  is  a  sine 
wave  of  double  frequency  about  an  axis  whose  distance  from 
the  zero  line  is  the  distance  representing  the  average  power, 
El  cos  </>. 

In  case  (1)  cos  0  is  unity  and  the  ordinate  of  the  axis  of 
the  power  wave  is  EL  The  maximum  power  is  2EI  and 
the  minimum  is  zero.  The  instantaneous  power  is  positive 
at  all  tunes;  that  is,  the  transfer  of  power  is  always  hi  the 


SINE  WAVE  ALTERNATING  CURRENTS 


137 


same  direction,  namely  from  the  source  of  power  into  the 
circuit. 

In  case  (2)  the  ordinate  of  the  axis  of  the  power  wave  is 
El  cos  0.  The  maximum  power  is  El  cos  <f>+EL  The 
minimum  power  is  El  cos  4>—EI  and  is  evidently  negative; 


FIG.  72. 

that  is,  during  such  times  as  the  current  and  e.m.f.  are  in 
opposite  directions,  the  flow  of  power  is  from  the  circuit 
back  to  the  source.  This  means  that  during  such  times  the 
energy  of  the  circuit  is  being  supplied  by  the  magnetic  field 
associated  with  the  circuit. 


FIG.  73. 

In  case  (3)  cos  0  is  zero,  and  the  power  wave  axis  is  coin- 
cident with  the  zero  line.  The  maximum  power  is  El  and 
the  minimum  power  is  —El',  that  is,  the  average  power  is 
zero.  During  one-half  cycle  all  the  energy  goes  to  building 
up  a  magnetic  field  and  during  the  next  half  cycle  all  that 
energy  is  returned  by  the  collapsing  field. 


138  ELECTRIC  AND  MAGNETIC  CIRCUITS 

If  the  circuit  contained  resistance  and  capacity,  the 
curves  would  be  similar  to  those  of  case  (2),  but  the  current 
and  power  waves  would  be  shifted  along  the  time  axis.  A 
similar  statement  applies  to  a  circuit  of  capacity  alone  with 
reference  to  case  (3). 

94.  Power  Factor.  Apparent  Power.  Reactive  Factor.— 
The  ratio  of  the  average  power  developed  in  a  circuit  to 
the  product  of  the  effective  values  of  e.m.f.  and  current  is 
defined  as  the  power  factor  of  the  circuit.  The  product 
just  mentioned  is  generally  called  the  volt-amperes  (or 
kilo  volt-amperes,  abbreviated  kv-a.}  of  the  circuit.  The 
power  (the  word  is  generally  understood  to  mean  average 
power,  unless  otherwise  specified)  has  just  been  proven  to 
be  equal  to  El  cos  <j>.  Its  ratio  to  the  volt-amperes  (El) 
is  evidently  cos  <£;  that  is,  power  factor  is  equal  to  cos  ^ 
for  the  case  under  discussion,  namely,  sine  waves  of  both 
e.m.f.  and  current.  It  should  be  particularly  noted  that  the 
power  factor  is  not  defined  as  cos  <t>  but  it  is  equal  to  cos  </> 
in  the  case  of,  and  only  in  the  case  of,  sine  waves  of  both 
e.m.f.  and  current.  Under  these  conditions,  <£  is  frequently 
called  the  power  factor  angle  of  the  circuit. 

The  product  of  the  e.m.f.  and  current  is  sometimes  called 
the  apparent  power.  The  rating  of  an  electrical  machine  is 
based  upon  its  apparent  power  rather  than  upon  its  real 
power  for  the  following  reasons :  Primarily  the  rating  depends 
upon  the  temperature  rise,  which  in  turn  depends  upon  the 
losses  in  the  machine.  The  losses  which  affect  the  rating  are 
of  two  kinds:  1st,  those  caused  by  hysteresis  and  eddy  cur- 
rents, and  2d,  those  caused  by  the  flow  of  current  through  the 
windings  of  the  machine.  The  hysteresis  and  eddy  current 
losses  are  a  function  of  the  magnetic  flux  and  consequently 
of  the  voltage  generated;  the  resistance  losses  are  a  func- 
tion of  the  current  which  flows.  A  given  voltage  and  a 
given  current  will  cause  the  same  total  loss  and  consequently 
produce  the  same  temperature  rise  whether  they  are  in 
phase  with  each  other  or  not.  Therefore,  the  product  of 
these  two,  rather  than  the  actual  power  developed,  deter- 


SINE  WAVE  ALTERNATING  CURRENTS 


139 


mines  the  temperature  rise  of  the  machine.  On  this  account, 
alternators  and  transformers  are  commonly  rated  in  kilo- 
volt-amperes,  or  KVA,  instead  of  in  kilowatts,  or  KW. 

The  product  of  the  volt-amperes  and  the  sine  of  the  angle 
of  phase  difference  (El  sin  $),  is  called  the  reactive  power 
of  the  circuit,  and  sin  </>  is  called  the  reactive  factor.  From 
the  relation  sin2  <£+cos2  <£  =  !,  we  get 


Reactive  Factor  =  Vl-  (Power  Factor)2.        (188) 

95.  Power  and  Reactive   Components   of  E.M.F. — In 

the  vector  diagram,  Fig.  74,  the  e.m.f .  E  may,  in  accordance 
with  the  principles  explained  in  Article  90,  be  considered  as 


L 

FIG.  74. 


the  vector  sum  of  two  e.m.f 's  (E  cos  0)  and  (E  sin  0),  the 
former  being  in  phase  with  I  and  the  latter  being  90°  ahead 
of  7.  This  resolution  of  E  into  two  components  is  expressed 
mathematically  by  the  equation : 


=  V(E  cos 


sin  0) 


(189) 


The  part,  E  cos  0,  is  called  the  power  component  of  the 
e.m.f.  because  the  product  of  it  and  the  current  gives  the 
power  developed  in  the  circuit.  The  part,  E  sin  0,  is 
called  the  reactive  component  of  the  e.m.f.  It  has  already 
been  proven  (see  equation  175)  that  the  total  e.m.f.  required 
for  an  alternating  current  circuit  containing  resistance  and 
reactance  is  E  =  V(RI)2+(XI)2,  where  RI  is  in  phase 


140 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


with  I,  and  XI  is  90°  ahead  of  /.  It  is  therefore  evident 
that 

Ecos<j>  =  RI,  (190) 

and  that 

Esm  <f>=XL  (191) 

The  power  component  of  the  e.m.f .  is  therefore  that  part  of 
the  e.m.f.  required  to  overcome  the  resistance  of  the  circuit, 
and  the  reactive  component  is  that  part  required  to  overcome 
the  reactance  of  the  circuit. 

From  equations  (190)  and  (180)  we  get  the  relation  that 


cos  <t>  = 


(192) 


that  is,  the  power  factor  of  a  circuit  is  equal  to  its  resistance 
divided  by  its  impedance. 

From  equations  (191)  and  (180)  we  get  the  relation  that 


sin  <t>=XI/E  =  X/Z,  (193) 

that  is,  the  reactive  factor  of  a  circuit  is  equal  to  its  reac- 
tance divided  by  its  impedance.  Dividing  equation  (193) 
by  equation  (192)  gives 

tan  $=X/R. 

96.  Power  and  Reactive  Components  of  Current;  Con- 
ductance, Susceptance  and  Admittance. — In  the  vector 
diagram,  Fig.  75,  the  current  is  considered  as  the  vector 

7cos0(^ 


I(-YE) 


FIG.  75. 


sum  of  two  currents,  (/  cos  </>)  and  (7  sin  #),  the  former 
being  in  phase  with  E  and  the  latter  90°  behind  E.  The 
part  I  cos  $  is  called  the  power  component  of  the  current 


SINE  WAVE  ALTERNATING  CURRENTS          141 

and  the  part  /  sin  0  is  called  the  reactive  component  of  the 
current.  The  resultant  of  these  two  components  is,  of 
course,  equal  to  /,  or 


7  =  V(I  cos  0)2+  (/  sin  0)2.  (194) 

The  values  of  7  cos  </>  and  7  sin  0  may  be  obtained  in  terms 
of  Ej  R,  and  Z  as  follows : 

7  cos  0  -  (E/Z)  (R/Z)  =  E(R/Z2)  =GE.          (195) 

where  G  is  a  symbol  for  the  expression  (R/Z2)  and  is    called 
the  conductance  of  the  circuit. 

7  sin  0  -  (E/Z)  (X/Z)  =  E(X/Z2)  =  BE,          (196) 

where  B  is  a  symbol  for  the  expression  (X/Z2)  and  is  called 
the  susceptance  of  the  circuit. 

Substituting  (GE)  and  (BE)  for  (7  cos  <f>)  and  (7  sin  </>) 
in  equation  (194)  we  obtain 


I  =  V(GE)2+(BE)2=EVG2+B2.  (197) 

Since  7  has  already  been  proven  equal  to  E/Z,  it  follows 
that 

Y}  (198) 


where  Y  is  a  symbol  for  the  reciprocal  of  Z  and  is  called  the 
admittance  of  the  circuit. 

From  the  relation  that  G  =  R/Z2,  it  follows  that 

R=GZ2=G/Y2  (199) 

and  from  the  relation  that  B  =X/Z2,  it  follows  that 

(200) 


97.  The  Symbolic  Method  of  Expressing  Vector  Quan- 
tities. —  The  manner  of  writing  vector  quantities,  the  cal- 
culation of  combinations  of  vector  quantities,  and  the 
interpretation  of  resulting  vector  quantities  may  be  much 
simplified  by  the  following  simple  convention:  Let  all  vec- 
tors be  considered  as  composed  of  two  component  vectors, 


142  ELECTRIC  AND  MAGNETIC  CIRCUITS 

one  along  any  chosen  axis  of  reference  and  the  other  at 
right  angles  to  the  chosen  axis;  and  let  the  component  at 
right  angles  to  the  axis  of  reference  be  designated  by  affixing 
to  it  the  symbol  j.  Thus,  the  vector  E  in  Fig.  74,  would  be 
written 

E  =  Ejcos  0  +jE  sin  0.  (201) 

and  the  vector  7  in  Fig.  75,  would  be  written 

I  =  1  cos  4>  -jl  sin  0.  (202) 

In  the  first  case,  the  axis  of  reference  is  the  vector  7  which 
would  be  written  7  =  7  —  j'O,  and  in  the  second  case  the  axis 
of  reference  is  the  vector  E}  which  would  be  written 


The  algebraic  sign  in  front  of  j  indicates  whether  the 
component  to  which  it  is  affixed  is  90°  ahead  of  or  90° 
behind  the  reference  axis;  the  plus  (+)  sign  is  used  when  it 
it  is  ahead  and  the  minus  (  —  )  sign  when  it  is  behind.  In 
an  algebraic  sense  the  right  angle  component  is  multiplied 
by  j  and  we  may  therefore  say  that  multiplying  a  vector 
by  j  rotates  it  90°  from  the  position  it  would  occupy  if  not 
multiplied  by  j.  If  a  right  angle  vector,  such  as  jE  sin  </>, 
be  again  multiplied  by  j,  it  becomes  j2E  sin  <£,  and  is  rotated 
another  90°,  or  180°  altogether;  but  it  is  then  equal  to 
—  E  sin  0  and  we  may  write  the  equation 

j2E  sin  (f>=  -E  sin  <j>, 
or 

J2=-l, 
or 

J-V-1,  (203) 

and  thus  j  is  seen  to  be  mathematically  equal  to  the  so-called 
imaginary  quantity,  V  —  1. 

Without  giving  the  mathematical  proof,  it  may  be 
stated  that  when  applied  to  vectors  in  one  plane  the  symbol  j 
obeys  all  the  laws  of  ordinary  algebra,  while  retaining  the 
special  significance  already  explained.  It  must  be  remem- 
bered, however,  that  an  equation  containing  the  symbol  j 


SINE  WAVE  ALTERNATING  CURRENTS         143 

is  a  special  vector  equation,  and  its  terms  cannot  be  trans- 
posed from  one  side  of  the  equality  sign  to  the  other.  It 
should  also  be  remembered  that  j2  may  always  be  replaced 
by  -1. 

The  numerical  value  of  any  vector  expressed  in  symbolic 
notation  must  always  be  found  by  extracting  the  square  root 
of  the  sum  of  the  squares  of  (the  sum  of  the  terms  not  con- 
taining j)  and  (the  sum  of  the  terms  containing/). 

98.  Impedance  and  Admittance  as  Complex  Numbers.— 
It  has  been  shown  that  the  e.m.f .  required  to  send  a  current 
/  through  a  resistance  R  and  a  reactance  X  is 


Written  in  symbolic  notation,  this  becomes 

E  =  RI+jXI  =  (R+jX)I  =ZL  (204) 

Thus  the  symbolic  expression  for  an  impedance  is  R+jX, 
and  expressed  in  this  form  it  is  called  a  complex  number. 
The  absolute  value  of  the  number  is  not  the  algebraic  sum 
of  its  two  parts  but  is  equal  to  the  square  root  of  the  sum  of 
the  squares  of  its  two  parts.  The  three  factors,  R,  X,  and 
Z  are  thus  related  to  each  other  as  the  three  sides  of  a  right- 
angled  triangle  with  Z  as  the  hypothenuse.  Such  a  triangle 
is  called  an  impedance  triangle.  Note  that  R,  X,  and  Z 
are  not  vectors,  but  are  simple  numerics,  although  the 
symbolic  expression  for  Z  is  of  the  same  mathematical 
form  as  that  of  a  vector.  The  algebraic  sign  of  an  inductive 
reactance  is  positive  and  of  a  capacity  reactance  is  negative. 
Thus,  if  R  =4  ohms  and  X  =  3  ohms  (the  plus  sign  is  under- 
stood), the  e.m.f.  required  to  send  20  amperes  through  the 
circuit  will  be 

E  =  20(4  +J3)  =  80  +J60  =  100,  (205) 

and  the  fact  that  E  leads  I  is  indicated  by  the  plus  sign 
before  j"60.  If  X  =  -3,  the  e.m.f.  will  be 

E  =  20(4  -  J3)  =  80  -  j60  =  100,  (206) 


144  ELECTRIC  AND  MAGNETIC  CIRCUITS 

and  the  fact  that  E  lags  behind  /  is  indicated  by  the  minus 
sign  before  ^60. 

In  a  similar  manner,  the  admittance  of  a  circuit  is  sym- 
bolically expressed  as  G  -jB.  The  symbolic  expression  for 
equation  (197)  is 

7  =GE  -jBE  =  (G  -JB}E  =  YE,  (207) 

the  minus  sign  being  used  here  because,  if  E  leads,  then  7 
lags,  and  if  a  positive  sign  is  used  to  indicate  a  leading  vector, 
than  a  negative  sign  must  be  used  to  indicate  a  lagging  vec- 
tor. The  algebraic  sign  of  an  inductive  susceptance  is  pos- 
itive (same  as  inductive  reactance)  and  of  a  capacity  sus- 
ceptance is  negative  (same  as  capacity  reactance).  Thus 
if  R  =4,  X  =  3,  Z  =  \/42+32=5,  then  G  =  4/25  =0.16  (see 
equation  195),  and  B  =3/25  =0.12  (see  equation  196). 
The  current  that  will  flow  under  an  impressed  e.m.f.  of 
E  =  100  will  be 

7  =  100  (0.16-j0.12)=16-jl2=20,  (208) 

and  the  fact  that  7  lags  is  indicated  by  the  minus  sign  before 
jl2.  If  X=  -3,  then  B  =  -0.12,  and  the  current  will  be 

7  =  100(0.  16  +jO.  12)  =16+jl2  =  20,  (209) 

and  the  fact  that  7  leads  is  indicated  by  the  plus  sign  before 
jl2.  Thus  equations  (205)  and  (208)  indicate  the  same 
phase  relation  between  E  and  7,  the  former  showing  that 
E  leads  7,  and  latter  showing  that  7  lags  behind  E.  Like- 
wise equations  (206)  and  (209)  indicate  that  E  lags  behind 
7,  or  that  7  leads  E. 

99.  Impedances  in  Series.  —  If  two  or  more  impedances 
are  in  series,  the  total  e.m.f.  is  equal  to  the  vector  sum  of  the 
e.m.f  s  required  for  the  separate  impedances.  Thus,  if  two 
impedances,  Ri+jXi=Zi,  and  R2+jX2=Z2,  are  in  series, 
the  e.m.f.  on  Z\  is 


and  on  £2  it  is 

E2  =  I(R2  +JX2)  =  77? 


SINE  WAVE  ALTERNATING  CURRENTS 


145 


but  IR  i  will  be  in  phase  with  IR2  and  thus  may  be  added 
directly;  and  IX  \  is  in  phase  with  (or  in  opposition  to)  IX2 
and  may  also  be  added  directly  (algebraically).  The  total 
e.m.f.  is  therefore 

I=ZQI,  (210) 


where  Ro,  Xo,  and  ZQ  are  the  equivalent  resistance,  reac- 
tance, and  impedance,  respectively,  of  the  whole  circuit. 
The  equivalent  impedance  of  a  series  circuit  is  therefore 

Z0  =  (R1+R2  +  .  .  .  )+j(Xl+X2  +  ...)=R0+jXo,      (211) 

that  is,  the  equivalent  resistance  of  a  series  circuit  is  the 
sum  of  the  separate  resistances,  the  equivalent  reactance  is 
the  sum  (algebraic)  of  the  separate  reactances,  and  the  equiva- 
lent impedance  is  the  square  root  of  the  sum  of  the  squares 
of  the  equivalent  resistance  and  equivalent  reactance. 

Note  particularly  that  the  total  impedance  is  not  the 
arithmetical  sum  of  the  separate  impedances,  but  can  be 

6 


FIG.  76. 

calculated  only  when  the  separate  resistances  and  reac- 
tances are  known.  Note  also  that  the  total  e.m.f.  is  not  the 
arithmetical  sum  of  the  e.m.f  s  on  the  separate  impedances, 
unless  the  separate  e.m.f  s  are  in  phase  with  each  other.  A 
study  of  Fig.  76  should  make  this  clear.  If  Xi/Ri  =X2/R2 
then  </>i  =  02  and  EI  will  be  in  phase  with  E2  and  E0  =  E±+E2 
but  otherwise  and  generally  EQ  must  be  calculated  by 
equation  (210). 

100.  Electromotive  Forces  in  Series. — In  most  practical 


146  ELECTRIC  AND  MAGNETIC  CIRCUITS 

problems,  the  circuits  consist  not  only  of  resistance  and 
reactance,  but  also  of  power-consuming  devices  other  than 
resistance,  such  as  motors.  Of  course,  a  motor  can  be 
represented  by  equivalent  values  of  resistance  and  reactance, 
the  resistance  being  of  such  value  that  when  multiplied  by 
the  current  taken  by  the  motor  it  gives  the  value  of  the  active 
component,  E  cos  <£,  of  the  voltage  impressed  on  the  motor, 
and  the  reactance  being  of  such  a  value  that  when  mul- 
tiplied by  the  current  it  gives  the  value  of  the  reactive  com- 
ponent, E  sin  </>.  However,  it  is  generally  unnecessary  to 
calculate  these  equivalent  values  of  resistance  and  reac- 
tance. The  data  generally  given  for  power  consuming 
devices  other  than  resistance,  are  the  E.M.F.,  Current,  and 
Power,  or  the  E.M.F.,  Power,  and  Power  Factor,  or  the 
E.M.F.,  Current,  and  Power  Factor.  In  series  circuits 
consisting  of  resistances,  reactances,  and  power-consuming 
devices  other  than  resistance,  the  total  active  E.M.F. 
will  be  RI  +  E  cos  0,  and  the  total  reactive  E.M.F.  will  be 
XI +E  sin  <£.  For  example,  consider  the  circuit  represented 
by  Fig.  77,  in  which  the  impedance  at  the  right  represents 


*-l 


FIG.  77. 

the  load  at  the  end  of  a  single  phase  transmission  line  and  Z 
is  the  impedance  of  the  line  itself.  The  equation  for  the 
voltage  at  the  station  is 

Eo  =  (Ei  cos  4>i  +RI)  +j(Ei  sin  0i  +XI),          (212) 

or 

E0  =  V(Ei  cos  4>i+RI)2  +  (Ei  sin  0i+X/)2,      (213) 

and  the  vector  diagram  is  shown  in  Fig.  78.  The  triangle 
Oab  is  the  one  represented  by  equation  (213). 

Equation  (213)  gives  the  best  form  for  the  calculation  of 
Eo  when  EI  is  given;  but  it  frequently  happens  that  E0  is 


SINE  WAVE  ALTERNATING  CURRENTS         147 


given  and  E\  must  be  found.  This  is  a  case  of  subtracting 
vectorially  the  impedance  drop  in  Z  from  the  e.m.f.  E0. 
An  inspection  of  the  diagram  shows  that  if  a  right  triangle 
be  constructed  on  E\  extended  to  the  point  c, 

EQ  =  (Ei  +  RI  cos  0i  +XI  sin  0i) 

+j(XI  cos  <f>i-RI  sin  0i).     (214) 

The  solution  of  this  equation  for  EI  gives 


i  =  VE0?  -  (XI  cos  0i  -RI  sin  0i)2 

-  (RI  cos 


+X7sin  0i).     (215) 


FIG.  78. 


This  equation  (215)  may  also  be  readily  'deduced  directly 
from  the  vector  diagram. 

The  power  factor  at  the  station  is 


COS  0o  = 


cos  <{>i+RI 


(216) 


101.  Resonance  in  Series  Circuits. — In  a  series  circuit, 
Fig.  79a  if  R  is  the  total  resistance,  L  is  the  total  induc- 
tance, and  C  is  the  total  capacity,  then  the  total  impedance  is 
(see  equation  176) 


(217) 
capacity 


where  Xi=the  inductive  reactance  and  ^2 
reactance.     Suppose  a  certain  circuit   has  E= 


148 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


and  X2  =  12;    the  negative  sign  for  X2  is   already  written 
into  equation  (217).     The  impedance  will  be 


Z=3+j(16-12)=3+.74=5, 


(218) 


and  if  an  e.m.f.  of  100  volts  is  impressed  on  the  circuit,  the 
current  will  be 

7  =  100/5=20,  (219) 


R 


O) 


(O 


FIG.  79. 

and  the  symbolic  equation  for  the  e.m.f.  will  be 

E  =  20[3  +j(16  - 12)]  =  60  +^(320  -240) 

=  60 +,;80  =  100.     (220) 

The  vector  diagram  for  the  circuit  is  shown  in  Fig.  79  b. 
Note  that  the  e.m.f.  across  the  inductive  reactance  is  320 
volts,  and  that  across  the  capacity  is  240,  both  of  which 
are  higher  than  that  across  the  entire  circuit.  .  If  the  capac- 
ity reactance  is  increased  to  16  ohms,  then  the  impedance 


SINE  WAVE  ALTERNATING  CURRENTS         149 

becomes  Z=3+jO=3,  and  7  =  100/3=33.3  while  the  volt- 
age across  each  reactance  will  rise  to  33.3x16=533  volts. 
This  condition  of  equality  between  the  inductive  reactance 
and  the  capacity  reactance  is  known  as  resonance,  and  in 
general  is  to  be  guarded  against,  lest  the  voltage  across  the 
inductance  or  the  capacity  or  both,  shall  rise  to  a  dangerous 
value. 

An  inspection  of  equation  (217)  shows  that  with  con- 
stant values  of  Rj  L,  and  C,  the  impedance  varies  with  the 
frequency  of  the  circuit  and  will  be  a  minimum  when 
C  or  when 


(221) 

This  value  of  frequency  is  known  as  the  resonant  frequency. 
It  is  worth  noting  that  the  current  in  such  a  circuit,  and  with 
it  the  potential  across  the  inductance  and  across  the  capacity, 
may  rise  very  rapidly  as  the  frequency  approaches  its  reso- 
nant value.  This  is  especially  so  when  R  is  small  in  com- 
parison with  Xi  and  X2  at  resonant  frequency. 

102.  Impedances  in  Parallel.  —  If  two  or  more  impedances 
are  in  parallel,  the  total  current  is  equal  to  the  vector  sum 
of  the  separate  currents.  Thus,  if  two  impedances, 
#1  +jXi  =Zi,  and  R2+JX2,  =  Z2)  are  in  parallel,  the  current 
in  Zi  is  Ii  =  E(Gi  -jBi)  and  in  Z2  is  I2=E(G2-jB2).  But 
EG  i  and  EG  2  are  the  two  power  components  of  the  current, 
are  therefore  in  phase  with  each  other  and  may  be  added 
directly;  EBi  and  EB2  are  the  two  reactive  components  of 
the  current  and  these  may  also  be  added  directly  (alge- 
braically). The  total  current  is  therefore, 

I  =  E(Gi  +G2)  -JE(B1+B2)  =  (G  -jB)E  =  YE,     (222) 

where  G,  B,  and  Y  are  the  equivalent  conductance,  suscep- 
tance  and  admittance,  respectively,  of  the  whole  circuit. 
The  equivalent  admittance  may  therefore  be  written  as 

Y  =  (G,+G2+  .  .  .  )  -j(J3i+B2  +  .  .  .  )  =G-jB,  (223) 
and  the  equivalent  impedance  of  the  whole  circuit  is 

Z  =  (G/Y2)  +j(B/Y2)  =  R  +JX,  (224) 


150 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


where  G/Y2  =  R,  the  equivalent  resistance  of  the  parallel 
circuit,  and  B/Y2=X,  the  equivalent  reactance  of  the  par- 
allel circuit.  Note  especially  that  the  resistance  is  not  the 
reciprocal  of  conductance,  nor  reactance  the  reciprocal  of 
susceptance,  but  that  impedance  is  the  reciprocal  of  admit- 
tance. Note  also  that  the  currents  can  not  be  added 
arithmetically,  but  must  be  added  vectorially;  and  that  the 
separate  admittances  cannot  be  added  arithmetically  but 
must  be  combined  as  indicated  in  equation  (223). 

103.  Currents  in  Parallel.— In  Fig.  80,  let  Zl  and  Z2 
represent  two  impedances  or  other  loads  in  parallel,  and  let 


t 

aoofiaaiP  -r  — 

0 

Q 

1 

•*  o 

0 

Zi 

0 

f 

i 

E 

1 

0 

0 

0 
0 
0 

o 

0 

'! 

0 
0 

1 

is, 


FIG.  80. 

the  currents  be  I\  and  1 2  respectively.  It  is  not  necessarj^  to 
calculate  the  admittance  of  the  combined  circuit;  the  total 
current  can  be  found  most  readily  by  calculating  the  active 
and  reactive  components  of  each  current  and  combining 
them  by  the  equation, 

7o  =  (/i  cos  0i  +72  cos  02)  -j(Ii  sin  0i  +  /2  sin  02),  (225) 
or 


cos 


cos 


sn 


sn 


The  resultant  power  factor  is 

COS  00  =  (/I  COS  01+/2  COS  $2)  //O. 


.  (226) 


(227) 


The  total  current  lags  if  (I\  sin  </>i  +/2  sin  <£2)  is  positive  and 
leads  if  it  is  negative.  Fig.  81  (a)  snows  the  vector  diagram 
for  a  case  where  <£i  is  positive  and  <£2  is  negative.  Fig.  81  (6) 
shows  the  diagram  for  a  case  where  <£i  and  02  are  both  posi- 
tive. 


SINE  WAVE  ALTERNATING  CURRENTS 


151 


104.  Mixed  Circuits. — In  calculations  on  combined 
series  and  parallel  circuits,  great  caution  must  be  exercised 
to  be  sure  that  in  adding  e.m.f  s  or  currents,  the  different 
components  are  expressed  in  terms  of  the  same  reference 


E 


FIG.  816. 


•E 


vector.  It  is  frequently  necessary  to  change  the  vector 
with  reference  to  which  a  current  or  e.m.f.  is  expressed.  A 
careful  study  of  the  following  problem  should  give  an  under- 
standing of  the  application  of  these  principles.  •  Let  Fig.  82 


T~ 

k 


I3=75 
leading 

cos  0  3=  0.5 


E= 


lagging 

:cos02=Sicos^1=0-87 

0.94 


It 


FlG.  82. 


represent  a  circuit  in  which  it  is  desired  to  find  the  values  of 
/,  /o,  EQ,  and  cos  4>o.  To  find  /  it  is  first  necessary  to 
resolve  I\  and  1 2  into  components  with  reference  to  E. 


152  ELECTRIC  AND  MAGNETIC  CIRCUITS 

From  the  data  given,  we  find  sin</>i  =0.493  and  sin  </> 2  =0.3412. 
Therefore 

7i=  80(0.87 -jO.493)    =  69.6-j39.44  (a) 

1 2  =85(0.94  -J0.3412)  =   79.9  -J29.00  (6) 


and  /  =  149.5  -j'68.44  =  164.4.         (c) 

The  components  of  /  in  equation  (c)  are  with  reference  the 
voltage  E.  To  find  EQ  we  must  add  to  E,  the  drop  in  Z 
due  to  7;  to  do  this  we  must  get  the  components  of  E  with 
respect  to  7,  and  add  RI  and  XI  to  the  active  and  reactive 
components,  respectively.  Letting  4>  represent  the  angle 
between  E  and  7,  we  get  cos  0  =  149.5/164.4=0.909  and 
sin  0=68.44/164.4=0.4164. 
Therefore,  with  respect  to  7 


E  =  1100     (0.909  +J0.4164)  =  1000  +J458  (d) 

ZI=   164.4  (0.8  +J0.7)  =   132+JH5  (e) 


and  #o  =  1132  +  J573  =  1269.    (/) 

Note  that  since  the  sign  of  j  is  negative  in  equation  (c)  it 
must  be  positive  in  equation  (d).  To  get  70,  we  must  add 
73  to  7;  from  the  data  given  we  get"  sin  </>3  =  —0.866;  there- 
fore 

73  =  75(0.5 +J0.866)  =  37.5+j64.95,  (g) 

with  respect  to  EQ.  Equation  (c)  gives  7  with  respect  to  E] 
therefore  the  components  of  7  in  equation  (c)  cannot  be 
added  to  those  of  73  in  equation  (g) ;  the  components  of  7 
must  be  found  with  respect  to  EQ.  Note  that  equation  (/) 
gives  EQ  with  respect  to  7;  therefore,  letting  0'  represent 
the  angle  between  EQ  and  7,  we  get 

cos  0' =  1132/1269  =0.892  and  sin  0'=  573/1269  =0.4515. 


SINE  WAVE  ALTERNATING  CURRENTS  153 
Therefore,  with  respect  to  E0 

I  =  164.4  (0.892  - J0.4515)  =  146.6  -j74.23  (h) 

I3=  75.0(0.5     +J0.866  )=  37.5+J68.95  (i) 


and  70  =  184.1  -j  5.28  =  184.2       (j) 

The  power  factor  of  the  combined  circuit  is  184.1/184.2  =  1.0 
(practically).  The  total  current  lags  the  total  voltage  by  a 
very  small  angle. 


CHAPTER  VII 
NON-HARMONIC  WAVES 

105.  Composition  of  Non-harmonic  Waves.  —  In  Chapter 
VI  our  attention  was  confined  entirely  to  the  phenomena  of 
sine  waves  and  the  methods  of  dealing  with  them.  Although 
hi  many  practical  engineering  problems  the  waves  are  close 
approximations  to  pure  sine  waves,  it  is  also  true  that  there 
are  many  cases  where  the  waves  depart  so  far  from  sinusoidal 
form  as  to  make  a  knowledge  of  the  mathematics  as  well  as 
of  the  physical  phenomena  of  non-harmonic  waves  a  neces- 
sity to  the  engineer. 

The  fundamental  proposition  in  dealing  with  non-har- 
monic waves  is  that  all  such  waves  may  be  represented 
mathematically  by  a  series  of  harmonic  terms,  called  Fou- 
rier's Series.  This  series  is  of  the  form 


El  sin  (ut+dl}+E2  sin 

sin  (3o>£  +  03).     (228) 


NOTE.  —  In  this  chapter,  a  numeric  following  a  letter  is  to  be  inter- 
preted as  a  subscript,  not  as  a  multiplier. 

Each  term  in  the  above  series  evidently  represents  a  sine 
wave.  The  first  term  is  called  the  first  harmonic,  or  fun- 
damental; its  maximum  value  is  El  and  the  angular  velocity 
of  the  vector  El  is  G>;  the  phase  relation  of  the  fundamental 
to  the  resultant  wave  is  represented  by  01,  the  origin  being 
taken  at  the  zero  value  of  the  original  or  resultant  wave, 
wlien  (•)£  =  0.  The  second  term  is  called  the  second  har- 
monic; its  maximum  value  is  E2,  its  angular  velocity  is 
twice  that  of  the  fundamental,  or  2w,  and  its  phase  relation 
to  the  resultant  wave  is  62.  Similarly,  the  third  term  is 

154 


NON-HARMONIC  WAVES  155 

called  the  third  harmonic  and  so  on.  In  plotting  a  wave 
and  its  components  it  must  be  noted  that  the  scale  of  angles 
for  any  harmonic,  say  the  nth,  is  n  times  the  scale  of  the  fun- 
damental, since  there  are  n  complete  harmonic  cycles  to  one 
fundamental  cycle.  62  and  03  as  used  in  the  above  equation 
are  expressed  in  terms  of  their  respective  harmonic  scales; 
for  example,  if  62  is  30°,  its  measure  on  the  scale  of  the 
fundamental  wave  would  be  15°. 

The  algebraic  sum  of  the  values  of  all  the  harmonics  and 
the  fundamental  at  any  instant  is  equal  to  the  correspond- 


FIG.  83. 

ing  instantaneous  value  of  the  original  or  resultant  wave. 
In  Fig.  83  is  shown  a  wave  (R)  whose  equation  is  e  =  160 
sin  <o£+40  sin  3(o£;  it  would  be  said  to  contain  a  25  per  cent 
third  harmonic.  In  Fig.  84,  the  equation  for  R  is  e  =  160 
sin  ut  +40  sin  2w£;  it  contains  a  25  per  cent  second  harmonic. 
It  will  be  noticed  that  in  Fig.  83  the  positive  and  negative 
halves  of  wave  R  are  alike,  while  in  Fig.  84  they  are  unlike. 
A  little  study  will  disclose  the  fact  that  any  even  harmonic 
will  produce  dissimilar  positive  and  negative  halves  in  the 
resultant  wave;  for,  the  values  of  the  fundamental  wave  at 
any  two  points  180°  apart  will  be  equal  and  opposite. in 
sign,  while  the  values  of  any  even  harmonic  at  two  points 
180°  apart  on  the  fundamental  scale  will  be  equal  and  alike 


156 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


in  sign;  the  sum  of  the  fundamental  and  the  harmonic  will 
therefore  be  different  at  points  180°  apart  (except  at  the 
points  where  they  pass  through  zero)  and  the  two  halves 
will  not  be  similar.  In  all  ordinary  electrical  machines,  the 
positive  and  negative  waves  are  similar,  and  it  follows  that 
even  harmonics  are  not  generally  to  be  found. 

In  actual  waves,  the  3d  and  5th  harmonics  are  generally 
the  ones  of  most  importance,  although  harmonics  of  a  higher 
order  are  by  no  means  uncommon  and  are  sometimes  of 


FIG.  84. 


considerable  magnitude.     Fig.  85  shows  a  wave  containing 
a  25  per  cent  3d  harmonic  and  a  16f  ^per  cent  5th  harmonic. 

106.  The  Oscillograph. — The  most  common  method  of 
determining  the  actual  form  of  an  alternating  e.m.f.  or  cur- 
rent wave  is  by  means  of  an  oscillograph.  This  instrument 
is  essentially  a  mirror  galvanometer  of  very  high  natural 
frequency,  so  as  to  be  able  to  follow  accurately  the  varia- 
tions in  the  wave  form.  For  voltage  waves  the  galvanom- 
eter coil  is  connected  in  series  with  a  high  non-inductive 
resistance,  as  in  the  case  of  a  voltmeter;  for  current  waves, 
the  coil  is  shunted  by  a  low  non-inductive  resistance.  If  a 
beam  of  light  is  thrown  onto  the  mirror  when  it  is  oscillating 
due  to  an  alternating  current  in  the  coil,  the  reflected  beam 


NON-HARMONIC  WAVES 


157 


will  also  oscillate  and  if  a  photographic  plate  or  film  be  moved 
with  sufficient  speed  across  the  path  of  this  reflected  beam 
and  at  right  angles  to  its  direction  of  motion,  the  position 
of  the  beam  will  be  photographed  in  all  its  positions  with 
respect  to  a  time  axis  and  this  will  constitute  a  photograph 
of  the  wave  form  of  the  current.  If,  instead  of  the  plate  or 
film,  a  second  mirror  be  placed  in  the  path  of  the  reflected 
beam  and  be  oscillated  about  an  axis  at  right  angles  to  the 


600 
500 
400 
300 
200 
100 
0 
100 
200 
300 
400 
500 
600 

f 

\ 

/ 

-x 

ts*~~ 

V, 

I 

1 
\ 

V 

// 

\ 

1 

\ 

\J 

-  —  Fundamental  Wave 

/ 

// 

^, 

\ 

\£—  Resultant  or  Original  Wave 

'/ 

3 

/ 

•d  Harmonic 

\ 

e  =  800«in(wfc  -f-10)-fl50sinC3a><+60;  — 

h 

/ 

5th  Harmonic 

/                     > 

/-^ 

\ 

Hs11 

WslnfSw 

t-t-3 

"/, 

0 

20A 

?^\ 

40 

59 

<$_*/ 

^ 

s 

16P/ 

180200\  2?/  2 

•oAN 

WU    300    ; 

20  3^ 

n 

V.  „ 

7 

\ 

\ 

J. 

\ 

X\ 

2 

V,  J 

/ 

\ 

/ 

n 

r 

s 

^ 

/ 

i 

V 

/ 

^ 

V 

\\ 

/ 

\ 

\\ 

s~ 

/ 

1 

S 

s^ 

/ 

\ 

/ 

/  1 

0306090120160180 
3rd  Harmonic  Scale 

^< 

^  

\ 

__.  •* 

7 

0    90     130 
5th  Harmonic  Scale 

V 

J 

FIG.  85. 


motion  of  the  beam  and  in  synchronism  with  that  motion, 
the  second  reflection  will  show  the  positions  of  the  first 
reflection  with  respect  to  a  time  axis  and  may  be  viewed 
upon  a  suitable  screen  as  a  standing  wave. 

107.  Analysis  of  a  Non-harmonic  Wave. — It  is  frequently 
desirable  to  know  the  magnitudes  and  phase  relations  of 
the  harmonic  components  of  a  non-harmonic  wave.  Having 
obtained  a  photograph  or  a  tracing  of  the  original  wave,  the 
analysis  involves  the  determination  of  the  maximum  values 
of  the  existing  harmonics,  the  magnitude  and  algebraic  signs 


158  ELECTRIC  AND  MAGNETIC  CIRCUITS 

of  the  phase  angles  of  the  harmonics  and  the  algebraic  signs 
of  the  harmonic  terms  as  a  whole.     That  is,  in  the  equation 


±E5  sin  (5o£±05)=h  etc.     (229) 

the  values  of  El,  E3,  E5,  el,  03,  05,  etc.,  must  be  found  and 
also  whether  the  algebraic  signs  are  plus  or  minus.  Several 
analytical  and  two  or  three  mechanical  methods  have  been 
devised  for  this  purpose.  The  method  to  be  explained  here 
is  one  of  the  simplest  analytical  methods  and  at  the  same 
tune  affords  a  good  insight  into  the  make-up  of  a  non- 
harmonic  wave. 

The  method  is  based  on  the  following  four  propositions: 
(1)  The  algebraic  sum  of  any  n  equally  spaced  ordinates 
of  a  sine  wave  is  zero  when  these  ordinates  are  so  spaced 
as  to  divide  k  complete  wave  lengths  into  n  equal  parts  and 
k  is  not  a  multiple  of  n.  For  example,  the  algebraic  sum  of 
3  ordinates  which  divide  1,  5,  or  7  wave  lengths  into  3  equal 
parts,  will  be  zero.  The  mathematical  proof  of  this  propo- 
sition will  not  be  given  here,  but  it  is  based  on  the  general 
principle  that  the  resultant  of  any  number  of  equal  vectors, 
equally  spaced  over  an  angle  of  360°,  is  zero.  These  may  be 
considered  as  independent  vectors,  or  as  different  positions 
of  the  same  vector  revolving  at  uniform  angular  velocity. 
The  vertical  projections  of  such  a  system  of  vector  positions, 
when  plotted  as  ordinates  against  the  corresponding  angular 
positions  of  the  vector  as  abscissae,  become  the  ordinates 
of  a  sine  wave,  and  the  sum  of  any  one  set  of  such  ordinates, 
corresponding  to  any  one  set  of  vector  positions,  equally 
spaced  over  360°,  is  zero.  To  illustrate,  in  Fig.  86  the 
sum  of  the  ordinates  uQ,  ul20  and  t/240  would  be  zero 
because  these  ordinates  are  the  projections  of  3  vectors 
(or  3  vector  positions)  equally  spaced  over  360°.  Similarly 
20+zl20+z240=0  for  the  same  reason.  Also,  ^0+^72  + 
w!44  +  u2  16  +7/288=0  because  these  ordinates  are  the 
projections  of  5  vector  positions  equally  spaced  over  360°; 
and  for  the  same  reason,  ZO  +£72  +Z144  +1216  +2288=  0. 


NON-HARMONIC  WAVES  159 

(2)  The  algebraic  sum  of  any  n  equally  spaced  ordinate 
of  a  sine  wave  is  equal  to  n  times  the  value  of  the  ordinates 
at  any  one  of  the  points,  when  these  ordinates  divide  k 
wave  lengths  into  n  equal  parts,  and  k  is  a  multiple  of  n. 
That  is,  the  ordinates  will  be  all  equal  and  of  the  same  sign. 
For  example,  3  ordinates  which  divide  3,  6,  or  9  wave  lengths 
into  3  equal  parts  will  have  a  sum  equal  to  3  times  the  value 
of  any  one  of  the  ordinates.  In  other  words,  if  the  number  of 
equally  spaced  ordinates  is  divisible  a  whole  number  of  times 
into  the  number  of  wave  lengths  used,  the  ordinates  will  be 
one  or  more  whole  wave  lengths  apart  and  their  sum  will  be 
n  times  the  value  of  any  one  of  the  ordinates.  For  example 
in  Fig.  86,' 

/O  +£120  -H240  =3£0; 


•  •  zO+z72+zl44+z216-f-2288=5zO. 

(3)  The  maximum  value  (E)  of  a  sine  wave  is  equal  to 
the  square  root  of  the  sum  of  the  squares  of  any  two  ordinates 
Oi)  and  (62)  90°  apart.     For  if 

ei  =E  sin  0  and  e2  =  E  sin  (0±90)  =E  cos  0, 
then 

ei2+e#  =  E2  (sin20+cos20)=#2. 

(4)  The  tangent  of  the    angle    between    any    ordinate 
(ei)  of  a  sine  wave  and  the  nearest  zero  point  on  the  wave  is 
equal  to  61/62  where  62  is  the  ordinate  of  the  wave,  90°  to 
the  right  of  61.     For  e\  =  E  sin  0  and  62  =  E  cos  0  and  there- 
fore ei/e2  =  E  sin  B/E  cos  0=tan  0.     If  e\  and  62  are  both 
positive,  the  wave  passes  through  zero  to  the  left  of  e\ 
and  upward;  if  e\  and  62  are  both  negative,  the  wave  passes 
through  zero  to  the  left  of  e\  and  downward;  if  61  is  positive 
and  62  is  negative  the  wave  passes  through  zero  to  the  right 
of  61  and  downward;  if  61  is  negative  and  62  is  positive,  the 
wave  passes  through  zero  to  the  right  of  e\  and  upward. 
The  algebraic  sign  of  the  numerical  expression  for  the  wave 
is  the  same  as  the  algebraic  sign  of  62.     We  have  then  the 
following  four  cases:  if  e\  and  62  are  positive,  the  wave  is 


160 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


represented  by  e  =  +E  sin  (ut+B) ;  if  e\  and  e2  are  negative, 
e  =  —  E  sin  (wZ  +  6) ;  if  e\  is  positive  and  e2  is  negative,  e  =  —  E 
sin  (co£-0);  if  ei  is  negative  and  e2  is  positive,  e  =  +E  sin 
(o>£-0). 

These  facts  regarding  the  relation  of  the  signs  of  the 
ordinates  and  the  position  of  the  wave  and  the  signs  in  the 
term  expressing  it,  can  be  most  easily  established  by  inspec- 
tion, and  the  student  should  make  such  an  inspection  of  the 
accompanying  curves. 


FIG.  86. 

Before  proceeding  with  the  application  of  these  prin- 
ciples to  the  analysis  of  a  wave,  the  following  fact  should  also 
be  carefully  noted:  All  ordinates  180°  apart  on  a  sine  wave 
are  equal  in  value  but  opposite  in  sign ;  therefore  if  n  ordinates 
be  equally  spaced  over  a  whole  number  of  half  wave  lengths, 
their  sum  will  be  the  same  as  if  spaced  over  an  equal  number 
of  whole  wave  lengths,  provided  the  algebraic  signs  of  the 
alternate  ordinates  be  reversed,  beginning  with  the  second 
ordinate.  For  example,  referring  to  Fig.  86,  uQ+ul20 
+^240=^0-^60+^120=0;  or,  again  zO+z72+z!44  + 
2216+2288  =zO  -286  +z72  -z!08  +2144  =5zO  and 


NON-HARMONIC  WAVES  161 

+1240=  tQ  -*60+tt20=3tf).  The  significance  of  this  fact 
is  that  only  one-half  of  a  complex  wave  need  be  drawn  for 
its  analysis. 

108.  Example.  —  On  Fig.  86,  let  the  curve  Y  be  a  non- 
harmonic  wave  which  has  been  found  by  the  oscillograph  or 
other  method  and  is  to  be  analyzed.  Briefly,  the  process 
is  to  determine  the  harmonics,  one  after  another,  then  add 
them  together  and  subtract  their  sum  from  the  original 
wave  to  determine  the  fundamental. 

The  formulae  given  below  are  general,  but  the  numerical 
values  for  the  accompanying  wave  are  added  for  illustra- 
tion. 

First,  take  three  ordinates,  yQ,  2/60,  and  7/120.  The 
value  of  the  third  harmonic  at  0°  is 

$  =  (2/0  -2/60+2/120)/3=  -13.4.  (230) 

(Note  that  2/0=0,  since  the  first  ordinate  is  taken  at  the 
origin.)  To  prove  equation  (230),  note  that  whatever  may 
be  the  values  of  the  fundamental  and  other  harmonics, 


also, 
and 


7/120=^120  +£120  +z!20, 


attention  being  given  to  algebraic  signs  in  each  case.     But 
from  propositions  (1)  and  (2) 


and 

tQ-  Z60+Z120  =3$. 
Therefore, 

2/0  -2/60+2/120=3^0. 

(If  the  wave  contained  a  9th  harmonic,  or  other  odd 
multiple  of  3,  it  also  would  be  included  in  the  sum  2/0—2/60 
+2/120.  To  determine  the  presence  or  absence  of  a  9th, 
for  example,  find  the  sum  of  9  ordinates  (2/0-2/20+  etc., 
up  to  2/160),  and  1/9  of  this  sum  is  nO,  the  value  of  the  9th 


162  ELECTRIC  AND  MAGNETIC  CIRCUITS 

harmonic  at  0°.  The  value  of  £0  is  then  found  by  subtract- 
ing nO  from  (yO  -y60  +y!20)  /3.  The  wave  shown  contains 
only  a  3d  and  5th. 

Second,  take  three  ordinates,  2/30,  2/90  and  7/150.  These 
ordinates  are  respectively  90°  (on  the  3d  harmonic  scale) 
to  the  right  of  yO,  2/60  and  1/120.  The  value  of  the  3d  har- 
monic at  30°  is,  by  the  same  reasoning  as  above, 

£30  =  (2/30  -2/90  +2/150)  /3  =  14.9.  (231) 

(If  the  wave  contained  a  9th  harmonic,  this  value  would 
also  have  to  be  corrected  in  the  same  manner  as  for  £0, 
namely  by  subtracting  (2/10—  2/30+  etc.,  to  t/170)/9.) 

Third,  by  proposition  (3),  the  maximum  value  of  the  3d 
harmonic  is 


#3  =  V(£0)2  +  (£30)2  =  20.  (232) 

Fourth,  by  proposition   (4),   the  tangent  of  the  angle 
between  the  origin  and  the  point  where  the  3d  harmonic 
crosses  the  X-axis  is 

tan  63  =  £0/£30  =  -13.4/14.9  =  -0.9.  (233) 

The  angle  is  therefore  —42°  (  —  14°  on  the  fundamental 
scale)  and  the  crossing  is  upward  14°  to  the  right  of  the 
origin,  since  £0  is  negative  and  £30  is  positive.  Also,  since 
£30  is  positive,  the  entire  third  harmonic  term  takes  the 
positive  sign;  that  is,  it  is  +20  sin  (3o>£  —  42).  This  is 
drawn  in  as  wave  T  on  the  diagram. 

Fifth,  take  five  ordinates,  2/0,  2/36,  y72,  2/108,  and  2/144. 
The  value  of  the  fifth  harmonic  at  0°  is  (if  no  multiple  of  the 
5th  is  present) 

*0  =  (  -2/36  +2/72  -2/108+2/144)/5=  -7.07.      (234) 

Sixth,  take  five  ordinates,  ylS,  2/54,  2/90,  2/126,  and  2/162. 
The  value  of  the  fifth  harmonic  at  18°  (which  on  the  5th 
harmonic  scale  is  90°  to  the  right  of  0)  is 

z!8  =  (2/18 -2/54 +2/90 -2/126+2/162)/5=  -7.07.      (235) 


NON-HARMONIC  WAVES  163 

Seventh,  find  the  maximum  value  of  the  fifth  harmonic, 


as 


E5  =  V(zO)2  +  (z!8)2  =  10.  (236) 

Eighth,  find  the  position  of  the  wave; 

tan  05.= zO /z!8  =  -7.07/- 7.07  =  1.  (237) 

The  angle  is  therefore  45°  (9°  on  the  fundamental  scale) 
and  since  zO  and  2! 8  are  both  negative,  the  nearest  crossing 
is  downward  9°  to  the  left  of  the  origin.  Since  z  18  is  nega- 
tive, the  expression  for  the  fifth  harmonic  is  — 10  sin  (5o>£ 
+45).  This  is  wave  Z  on  the  diagram. 

Ninth,  find  the  value  of  the  fundamental  at  0°  as 

^0  =  ?/0  -  /O  -  zO  =  20.45.  (238) 

Tenth,  find  the  value  of  the  fundamental  at  90°  as 

^90  =  ?/90  -£90  -z90  =7/90  +£30  -z!8  =50.7.       (239) 

Note  that  £90=  —  £30,  since  they  are  180°  apart  on  the 
3d  harmonic  scale;  also  that  z90=z!8,  since  they  are  360° 
apart  on  the  5th  harmonic  scale. 

Eleventh,  find  the  maximum  value  of  the  fundamental  as 


El  =  V(uO)2  +^902  =  54.6.  (240) 

Twelfth,  find  the  position  of  the  wave; 

tan  01  =uQ/u9Q  =20.45/50.6  =0.404.  (241) 

The  angle  is  therefore  22°  and  the  wave  crosses  the 
X-axis  upward  22°  to  the  left  of  0.  This  is  drawn  in  as 
wave  U  on  the  diagram. 

The  equation  for  the  original  wave  (F)  may  now  be 
written  as' 

e  =  54.6sin((D£+22)+20sin(3w£-42)-10sin(5w£+45).(242) 

If  harmonics  of  higher  order  than  those  mentioned  here 
are  present,  the  process  of  determining  them  is  the  same  as 


164  ELECTRIC  AND  MAGNETIC  CIRCUITS 

given  above  that  is,  for  a  7th  harmonic,  7  ordinates  would  be 
used,  25.7°  apart,  and  so  on.  In  each  case,  where  necessary, 
correction  must  be  made  for  the  higher  multiples  of  any 
harmonic,  as  explained  in  connection  with  the  3d  harmonic 
and  these  higher  multiples  must  also  be  completely  deter- 
mined and  included  in  the  final  equation  for  the  wave. 

109.  Effective  Value  of  a  Non-harmonic  Wave.—  The 
effective  value  of  an  alternating  wave  has  already  been  shown 
to  be  equal  to  the  square  root  of  the  mean  square  of  the 
instantaneous  values  taken  over  any  whole  number  of  cycles. 
If  the  equation  for  a  non-harmonic  wave  is 


=  El  sin  (<o«  +  01)  +#3  sin  (3w  +  03)  +#5  sin  (5w£  +  05),  (243) 
the  equation  for  the  curve  of  squared  values  is 


e2  =  El2  sin2  (co*  +01)  +E32  sin2  (3o>  +  03)  +E52  sin2 
+2E1  sin  (o>J  +  01)  E3  sin 
+2E3  sin  (3w^  +  03)  Eo  sin 
+2E1  sin  (co<  +  01)  E5  sin  (5^  +  05).     (244) 


The  mean  value  of  e2  is  equal  to  the  sum  of  the  average 
values  of  each  term  on  the  right-hand  side  of  equation  (244), 
each  average  being  taken  over  one  complete  cycle  of  fun- 
damental frequency.  Each  of  the  first  three  terms  of 
equation  (244)  may  be  expanded  into  the  form 


and  each  of  the  last  three  terms  may  be  expanded  in  the  form 
EmE»[cos  (z(ra-n)+0m-0w)-cos 


where  x  =  ut  and  m  and  n  are  the  order  of  the  respective 
harmonics.  Using  these  expansions,  multiplying  the  equa- 
tion by  dx,  integrating  between  the  limits  of  o  and  2?r, 
and  dividing  the  result  by  2?r,  we  get 

(average  e2  =  #12  +  E32  +  E&)  /2,  (245) 


NON-HARMONIC  WAVES  165 

and  the  effective  value  of  e  is 


(246) 

That  is,  the  effective  value  of  a  non-harmonic  e.m.f.  or  cur- 
rent is  equal  to  the  square  root  of  the  sum  of  the  squares  of 
the  effective  values  of  its  component  sine  waves,  or  equal 
to  .707  times  the  square  root  of  the  sum  of  the  squares  of 
the  maximum  values  of  its  component  sine  waves. 

110.  Peak  Factor. — The  ratio  of  the  maximum  value  of 
an  e.m.f.  or  current  to  its  effective  value  is  defined  as  its 
peakjactor,  or  crest  factor.     The  peak  factor  of  a  sine  wave 
is  \/2  =  1.414.     The  peak  factor  of  commercial  waves  may 
be  greater  or  less  than  this.     When  it  is  greater  the  wave  is 
called  a  peaked  wave  and  the  insulation  of  the  circuit  is 
subjected  to  a  greater  strain  than  with  a  sine  wave.     When 
the  peak  factor  is  less  than  1.414  the  wave  is  called  a  flat- 
topped  wave. 

111.  Average  Value  of  a  Non-harmonic  Wave. — The 
average  value  is  found  by  multiplying  equation  (243)  by 
d(o>0  and  integrating  between  the  limits  o  and  w  and  dividing 
the  result  by  ?r.  This  gives 

E  (average)  =.637  ( El  cos  01  +\-  cos  03+^  cos  05 ),    (247) 
\  o  o  / 

where  .637=  2  /V. 

112.  Form  Factor. — The  ratio  of  the  effective  value  of  an 
e.m.f.  or  current  to  its  average  value  is  defined  as  the  form 
factor  of  the  wave.     For  a  sine  wave  the  form  factor  is 
.707/.637  =  1.11. 

113.  Power  in  Circuits  Carrying  Non-harmonic  Waves. — 
Let  equation  (248)  be  the  e.m.f.  equation  for  a  circuit  and 
(249)  the  corresponding  current  equation: 

e=El  sin  (ci>£  +  01)+l£3  sin  (3wZ  +  03) 

+#5  sin  (5<o* +  05).     (248) 
i  =  71  sin  (o)£  +  01')+73  sin  (3o)£  +  03') 

+#5  sin  (5<o* +  05')  .     (249) 


166  ELECTRIC  AND  MAGNETIC  CIRCUITS 

The  instantaneous  power  in  the  circuit  will  be  the  product  of 
equations  (248)  and  (249).     This  product  is 


p=EUl  sin  (<*t  +  el)  sin 
+#373  sin  (3^  +  03)  sin  (3(^  +  03') 
+#575  sin  (5o>£  +  05)  sin  (5w£  +  05') 
+#173  sin  (otf  +  01)  sin  (3  CD*  +  03') 
+#371  sin  (arf  +  01')  sin  (3  to!  +  03) 
+#571  sin  (ut  +  er)  sin 
+#175  sin  (tit  +  01)  sin 
+#375  sin  (3wZ  +  03)  sin 
+#573  sin  (5o>Z  +  05)  sin  (3u>£+03').      (250) 


The  average  power  is  found  by  integrating  this  product 
multiplied  by  d(vt)  between  the  limits  o  and  2?r  and  dividing 
the  result  by  2?r;  that  is,  by  finding  the  sum  of  the  average 
values  of  each  term  over  a  complete  fundamental  cycle. 
By  the  same  process  as  that  used  in  Article  (109)  it  will  be 
found  that  the  average  value  of  each  of  the  last  six  terms  of 
the  equation  is  zero.  By  the  same  process  as  that  used  in 
Article  (93)  the  average  value  of  the  first  three  terms 
will  be  found  to  be 


#373  #575 

P  =  —  rr-  cos  01+—  g—  cos  03  +  —  2~  cos  05,      (251) 

where  01  =  01-01'  and  03  =  03-03'  and  05  =  05-05'. 
Note  that  01,  03,  and  05  are,  respectively,  the  phase  angles 
between  the  fundamental  e.m.f.  and  current,  the  3d  har- 
monic e.m.f.  and  current,  and  the  5th  harmonic  e.m.f.  and 
current.  Note  also  that  if  #1,  #3,  #5,  71,  73  and  75  be 
taken  to  represent  effective  values  instead  of  maximum 
values,  the  equation  for  power  becomes 

P  =  #171  cos  01  +#373  cos  03  +#575  cos  05.     (252) 

From  this  it  is  seen  that  the  total  average  power  is  equal  to 
the  sum  of  the  powers  which  would  be  produced  by  each 
component  e.m.f.  wave  and  its  corresponding  current  wave 
if  they  were  acting  independently  of  the  other  harmonics. 


NON-HARMONIC  WAVES  167 

114.  Equivalent  Sine  Waves  and  Phase  Difference.— 

Equation  (246)  gives  the  effective  value  for  the  non-harmonic 
wave  represented  by  equation  (243).  These  equations  hold, 
of  course,  for  either  an  e.m.f.  wave  or  a  current  wave.  If  E 
be  the  effective  value  of  a  non-harmonic  e.m.f.,  the  equa- 
tion for  the  sine  wave  which  would  be  exactly  equivalent 
to  the  non-harmonic  wave  is 

e  =  V2E  sin  <o«.  (253) 

If  /  be  the  effective  value  of  the  corresponding  current,  the 
equation  for  the  equivalent  sine  wave  of  current  is 

i  =  V2I  sin  (w/-0).  (254) 

The  power  in  the  circuit  is  represented  by  equation  (252) 
and  the  power  factor  is  P /El]  the  angle  <£  in  equation  (254) 
is  therefore  the  angle  whose  cosine  is  P  /El  and  is  called  the 
equivalent  angle  of  phase  difference.  It  bears  no  definite 
relation  to  any  of  the  angles  connected  with  the  non-har- 
monic waves,  but  is  the  angle  which  would  have  to  exist 
between  two  sine  waves  of  the  same  effective  values  as  the 
given  non-harmonic  waves,  in  order  that  the  power  shall 
be  the  same  as  that  given  by  the  non-harmonic  waves. 

115.  Inductive  Reactance  with  Non-harmonic  Wave.— 
The  inductive  reactance  due  to  an  inductance  L  has  been 
shown  to  be  equal  to  2irfL.     This  expression  applies  to  each 
harmonic  of  a  complex  wave,  when  /  is  the  frequency  of 
that  harmonic.     If/  is  the  fundamental  frequency,  the  fun- 
damental reactance  is  27T/L,  the  reactance  against  a  J3d 
harmonic  e.m.f.  will  be  67T/L,  and  against  a  5th  harmonic?  it 
will  be  10x/L.     Or,  if  the  reactance  to  the  fundamental  be 
called  #1  =  (27r/L),  then  the  3d  harmonic  reactance  is  3xi 
and  the  5th  harmonic  reactance  is  5#i.     These  facts  will  be 
evident  if  it  is  recalled  that  the  e.m.f.  to  overcome  inductance 
is  e  =  Ldi/dt'}    then  if  i  =  I  sin  2ir/£,  e  =  (2T/LV7  cos  2irft] 
\ii  =  I  sin  6ir/J,  e  =  (67r/L)7  cos  Qvft. 

It  follows  from  this,  that  if  a  circuit  contains  a  resistance 
r  and  a  reactance  x\  at  fundamental  frequency,  the  funda- 


168  ELECTRIC  AND  MAGNETIC  CIRCUITS 

mental  current  will  be  71  =  El/(r  +jxi),  the  3d  harmonic 
current  will  be  73  =E3/(r+j3xi)  and  the  5th  harmonic 
current  will  be  75  =  £r5/(r+j5^i). 

The  equation  of  the  resultant  effective  current  is  there- 
fore 


7m  __  _ 

'*2+*2+22' 


(See  equation  (246).) 

It  is  evident  that  the  ratio  of  73  to  71  will  be  less  than 
the  ratio  of  E3  and  El  and  the  ratio  of  75  to  71  will  be  still 
smaller.  That  is,  the  harmonic  currents  will  be  smaller  in 
proportion  to  the  fundamental  current  than  are  the  corre- 
sponding harmonic  e.m.f's  to  the  fundamental  e.m.f. 
Inductive  reactance  is  therefore  said  to  dampen  out  the 
harmonics  in  the  current  wave.  The  amount  of  damping 
will  of  course  depend  on  the  relative  values  of  r  and  x\. 
If  Xi  is  negligibly  small  there  will  be  no  damping;  if  r  is 
negligibly  small,  the  ration  of  the  3d  harmonic  current  to  its 
fundamental  will  be  1/3  of  the  ratio  of  3d  harmonic  e.m.f. 
to  its  fundamental. 

Furthermore,  the  phase  difference  between  the  component 
e.m.f.  and  current  waves  will  increase  with  the  order  of  the 
harmonic;  the  tangent  of  the  angle  of  lag  for  the  funda- 
mental will  be  xi/r;  for  the  3d  harmonic,  it  will  be  3zi/r; 
and  so  on.  Therefore  both  on  account  of  the  damping  and 
on  account  of  the  change  in  phase  relations,  the  shape  of 
the  current  wave  will  be  different  from  that  of  the  e.m.f. 
wave. 

A  method  frequently  used  for  determining  the  reactance 
(and  sometimes  the  inductance)  of  a  circuit  is  to  impress  a 
known  e.m.f.  on  the  circuit  and  measure  the  current.  This 
is  called  the  impedance  method.  The  ratio  of  the  e.m.f. 
to  the  current  is  the  impedance  (z)  of  the  circuit.  Then, 
the  resistance  of  the  circuit  having  been  measured,  the  reac- 
tance (x)  is  Vz2—  r2,  and  the  inductance  L  is  x/2-jrf,  where  / 
is  the  frequency  of  the  e.m.f.  The  use  of  this  expression 


NON-HARMONIC  WAVES  169 

assumes  the  e.m.f.  wave  to  be  sinusoidal,  and  it  should  be 
noted  that  this  value  of  L  will  not  be  correct  unless  the  e.m.f. 
wave  is  a  sine  wave,  and  also  noted  that  the  reactance  of  a 
given  coil  or  circuit  as  determined  for  one  wave  form  will 
not  be  the  same  for  any  other  wave  form.  This  can  be  most 
easily  shown  by  an  example:  Suppose  a  60-cycle  sine  wave 
of  100  volts  (effective)  be  impressed  on  a  circuit  of  negligible 
resistance  and  an  inductance  of  .01061  henry.  The  reac- 
tance at  60  cycles  will  be  2ir X60x. 01061  =377 X. 01061  = 
4  ohms.  The  effective  current  will  therefore  be  100/4=25 
amperes.  Now  suppose  the  e.m.f.  wave  is  non-harmonic 
with  a  fundamental  of  95.4  volts  (effective)  and  a  3d  har- 
monic of  30  volts  (effective).  The  effective  value  of  the 
wave  will  still  be  100  volts,  since  ^95. 42  +30*  =  100.  The 
fundamental  current  will  be  95.4/4=23.85  amperes  (effect- 
ive) and  the  3d  harmonic  current  will  be  30/(3x4)=2.5 
amperes  (effective).  The  total  effective  current  will  be 
V23.852+2.52=  23.98  amperes.  The  reactance  for  this 
wave  form  is  therefore  100/23.98=4.17  ohms,  and  if  the 
inductance  were  calculated  from  this  value  on  the  assump- 
tion that  the  wave  is  a  sine  wave,  the  result  would  be  4.17/ 
377  =  .01105,  which  is  in  error  by  over  4  per  cent. 

If  the  components  of  the  e.m.f.  wave  are  known,  the  value 
of  x\  could,  of  course,  be  calculated  from  equation  (255) 
but  this  is  obviously  a  rather  tedious  solution  and  is  imprac- 
tical. The  points  of  this  discussion  are  (1st)  that  it  is  unsafe 
to  assume  that  a  wave  is  sinusoidal  in  calculating  the  value 
of  inductance,  and  (2d)  that  the  reactance  of  a  given  circuit 
varies  with  the  wave  form  of  impressed  e.m.f. 

116.  Capacity  Reactance  with  Non-harmonic  Waves.— 
The  effect  of  capacity  reactance  with  non-harmonic  waves  is 
just  opposite  to  that  of  inductive  reactance.  Capacity 
reactance  has  been  shown  to  be  equal  to  1/27T/C.  This 
expression  applies  to  each  harmonic  of  a  complex  wave,  when 
/  is  the  frequency  of  that  harmonic.  If  /  is  the  fundamental 
frequency,  the  expression  as  given  is  the  fundamental 
reactance,  the  reactance  to  a  3d  harmonic  e.m.f.  is  1/67T/C, 


170  ELECTRIC  AND  MAGNETIC  CIRCUITS 

and  to  a  5th  harmonic  it  is  l/lOwfC.  If  the  fundamental 
reactance  be  called  x\,  the  3d  harmonic  reactance  is  xi/3 
and  the  5th  harmonic  reactance  is  xi/5.  The  fundamental 
current  will  be  El/xi]  the  3d  harmonic  current  will  be 
3E3/Xi,  and  the  5th  harmonic  current  will  be  5E5/Xi. 
From  this  it  is  evident  that  the  harmonic  currents  are 
increased  in  their  ratio  to  the  fundamental,  instead  of  dimin- 
ished as  in  the  case  of  inductance.  The  wave  form  of  the 
current  will  therefore  be  different  from  that  of  the  e.m.f. 
and  will  differ  more  widely  from  a  sine  wave  than  does  the 
e.m.f.  wave. 

The  relation  I  =2irfCE  may  be  used  to  determine  the 
capacity  of  a  condenser,  if  the  impressed  e.m.f.  wave  is 
sinusoidal,  but  will  give  incorrect  results  if  the  e.m.f.  wave 
is  non-harmonic.  The  formula  for  the  total  effective  current 
is 

I  =  V(2vfCEl?  +  (67r/C#3)2  +  (107r/C#5)2,         (256) 

and  this  formula  may  be  used  for  determining  C  if  the  com- 
ponents of  the  e.m.f.  wave  are  known. 

If  a  resistance  r,  and  inductive  reactance  x\  at  funda- 
mental frequency  and  a  capacity  reactance  x2  at  funda- 
mental frequency  are  in  series  the  fundamental  current  is 

Il=El/r+j(x1-x2),  (257) 

the  3d  harmonic  current  is 


73  =  E3/r+j(3xl  -x2/S),  (258) 

and  the  5th  harmonic  current  is 

z2/5).  (259) 


From  these  equations,  it  will  be  seen  that  the  phase  relations 
between  the  current  components  are  different  from  the 
phase  relations  between  the  e.m.f.  components,  that  the 
ratios  of  the  current  components  are  different  from  the 
ratios  of  the  e.m.f.  components,  and  therefore  the  current 
wave  form  will  be  different  from  the  e.m.f.  wave  form. 


CHAPTER  VIII 
POLYPHASE  CURRENTS 

117.  KirchhofFs  Laws  Applied  to  Alternating  Currents.— 

These  laws  as  stated  with  respect  to  direct  currents  of 
course  apply  also  to  instantaneous  values  of  alternating 
currents.  It  should  be  noted,  however,  that  these  laws  also 
apply  to  vector  quantities,  and  therefore  to  effective  values, 
provided  the  laws  are  so  stated  as  to  take  into  account  the 
vectorial  nature  of  the  quantities,  and  provided  that  in  apply- 
ing them,  all  vectors  are  resolved  into  their  components  with 
respect  to  some  one  vector  as  a  reference  line.  As  applying 
to  alternating  currents  and  e.m.f's,  the  laws  may  be  stated 
as  follows: 

I.  The  vector  sum  of  all  externally  induced  e.m.f's  in  a 
given  direction  around  a  closed  circuit  is  equal  to  the  vector 
sum  of  all  impedance  drops  in  the  same  direction  around  the 
circuit.     By  externally  induced  e.m.f's    are  meant  those 
e.m.f's  generated  by  relative  motion  between  inductors  and 
a  magnetic  field  where  the  field  is  not  produced  by  the  cur- 
rent  in   the   circuit   under   consideration.     A   self-induced 
e.m.f.  is  included  in  the  impedance  drop  as  that  part  of  the 
impedance  drop  which  is  consumed  as  reactance  drop. 

II.  The  vector  sum  of  all  currents  flowing  toward  a 
point  is  equal  to  the  vector  sum  of  all  currents  flowing  away 
from  the  point. 

In  both  these  laws,  where  direction  is  referred  to,  it 
must  be  understood  that  the  chosen  positive  direction  is 
meant,  and  that  all  phase  relations  must  be  expressed  with 
reference  to  these  chosen  positive  directions. 

118.  Two-phase   Connections. — In   Fig.   87   are   repre- 
sented the  essential  features  of  a  two-phase  four-pole  alter- 

171 


172 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


nator.  The  poles  are  represented  as  revolving,  as  is  usually 
the  case  with  alternators.  They  are  excited  by  direct  cur- 
rent, fed  into  the  field  coils  through  two  slip  rings.  The 
full  lines  connecting  the  armature  inductors  represent  the 
connections  at  the  front  end  of  the  armature  and  the  dotted 
lines  the  connections  at  the  rear  end.  The  terminals  of 
phase  (1)  are  a  and  6;  of  phase  2  are  c  and  d.  It  will  be 
noted  that  the  inductors  of  phase  (1)  are  90  electrical  degrees 
from  the  inductor  of  phase  (2);  therefore  the  e.m.fs  in 
the  two  phases  will  be  90°  out  of  phase.  Since  the  e.m.fs 
are  alternating  in  direction,  either  direction  may  be  thought 


FIG.  87. 


of  as  the  positive  direction,  and  it  is  therefore  impossible 
to  say  which  e.m.f.  is  ahead  of  the  other,  unless  one  direc- 
tion or  the  other  is  arbitrarily  chosen  as  the  positive  direc- 
tion. Let  the  arrows  marked  (1)  and  (2)  show  the  arbi- 
trarily chosen  positive  direction  in  phases  (1)  and  (2) 
respectively :  then  we  can  say  that,  with  the  particular  choice 
made,  the  e.m.f.  in  phase  (1)  is  90°  ahead  of  that  in  phase  (2). 
A  study  of  the  figure  will  show  that  the  e.m.f.  in  phase  (1) 
reaches  its  positive  maximum  90°  before  that  in  phase  (2) 
reaches  its  positive  maximum.  The  usual  convention  for 
representing  the  windings  of  an  alternator  is  shown  in  Fig. 
88,  Positive  directions  must  be  chosen  for  all  windings 


POLYPHASE  CURRENTS  173 

and  other  parts  of  the  circuits,  before  vector  relations  can  be 
be  expressed  either  graphically  or  mathematically.  These 
chosen  positive  directions  should  always  be  shown  by  prop- 
erly placed  arrows.  It  must  be  remembered  that  these  arrows 
do  not  represent  the  direction  of  e.m.f.  or  current  at  any 
particular  instant,  but  the  direction  which  is  considered  as 
the  positive  direction.  It  is  also  necessary  to  know  (or 
assume  to  be  known)  the  phase  relations  between  the 
induced  e.m.f  s  in  the  different  windings,  with  respect  to  the 
chosen  positive  directions.  For  example,  to  draw  the  vector 
diagram,  Fig.  89,  for  the  e.m.f  s  in  Fig.  88,  it  is  necessary 
to  know  that  E2  in  its  chosen  positive  direction  lags  90° 
behind  E\  in  its  chosen  positive  direction. 


0 


FIG.  88.  FIG.  89. 

If  the  two  circuits  of  a  two-phase  alternator  are  kept 
separate,  the  vector  diagrams  for  the  two  circuits  will  be 
distinct  single-phase  diagrams,  and  no  combined  diagram  is 
possible,  except  to  indicate  the  relative  positions  of  the 
vectors  as  in  Fig.  89.  It  is  usual,  however,  to  join  one  ter- 
minal of  each  phase  to  a  common  wire  in  the  case  of  a  two- 
phase  machine  and  use  three  wires  for  the  line  instead  of 
four,  as  shown  in  Fig.  90.  In  this  case,  with  the  positive 
directions  shown,  the  voltage  between  the  outside  wires  will 
be  the  vector  sum  of  the  two  e.m.f  s  and  the  current  in  the 
common  wire  will  be  the  vector  difference  between  the  two 
currents  in  the  outside  wires.  The  vector  diagram  is  shown 
in  Fig.  91.  Assuming  that  the  two  e.m.f  s  are  equal,  it  is 
seen  that  the  voltage  Vad  is  equal  to  ^2  times  the  phase 
e.m.f.  and  leads  E2  by  45°.  Mathematically  this  result 


174  ELECTRIC  AND  MAGNETIC  CIRCUITS 

may  be  found  as  follows:  Taking  E2  as  the  reference  vector, 
the  e.m.f.  E\  would  be  expressed  in  symbolic  notation 
as  jEi  ;  and  the  vector  sum  of  the  two  would  be 


=  E2  +JE1 


^  =  V2E2, 


(260) 


and  the  sine  of  the  angle  between  E2  and  Vad  is  .707;  there- 
fore Vad  leads  E2  by  45°. 

Assume  that  the  two  currents  are  equal  and  in  phase 
with  their  respective  e.m.f  s.      As  positive  direction  of  cur- 


'ad 


-*.  v 


ad 


FIG.  90. 


rent  in  the  common  wire  (ra)  choose  the  direction  indicated 
by  the  arrow.  Then,  since  the  positive  direction  of  /i 
is  away  from  the  junction  and  the  positive  direction  of  I2 
is  towards  the  junction,  Im  must  be  the  vector  difference  of 
1 2  and  Ii]  or 

/m  =  /2-j7i=V2/2,  (261) 

and  the  sine  of  the  angle  between  72  and  Im  is  —.707; 
therefore  Im  lags  45°  behind  72. 

Graphically,  subtracting  a  vector  means  that  it  is  to  be 
reversed  in  direction  and  then  added. 

When  a  receiving  circuit  is  fed  over  a  3-wire  2-phase  line 
as  shown  diagrammatically  in  Fig.  92,  the  system  becomes 
more  or  less  unbalanced  owing  to  the  effect  of  the  drop  in 
the  middle  wire.  This  is  shown  by  the  vector  diagram  in 


POLYPHASE  CURRENTS 


175 


Fig.  93.     The  load  currents,  h  and  76  are  assumed  to  be 
equal  and  to  lag  by  equal  angles  behind  the  generator  volt- 


ages 

^ 


— 'Tnnr&ffW'- 


La 


FIG.  92. 


FIG.  93. 


i  and  E2.      The  voltages  across  La  and  L&  are  respec- 

7a-  ^1-^4/4+^5/5,  262) 

(263) 


176  ELECTRIC  AND  MAGNETIC  CIRCUITS 

Note  that  the  drop  in  the  middle  wire  adds  vectorially  to  E\ 
because  the  positive  direction  chosen  in  it  is  opposite  to  that 
chosen  for  EI,  while  this  same  drop  subtracts  vectorially 
from  E2.  Observe  carefully  the  statements  concerning 
Kirchhoff's  Laws,  Article  117.  Equations  (262)  and  (263) 
are  vector  equations  and  all  quantities  must  be  resolved  into 
their  components  with  respect  to  some  one  vector  as  a 
reference  axis,  before  the  equations  can  be  solved  numer- 
ically. Using  EI  as  the  reference  vector,  the  current  /4 
must  be  expressed  as  /4  (cos0i  —  j  sin  0i)  and  75  as  75  (cos 
(01  +  135)—  j  sin  (01  +  135).  Substituting  these  values  in 
equation  (262)  and  expanding,  remembering  that  z±=r± 
and  Z5=r5+x5,  the  value  of  Va  is 


Va  =  EI  -r4/4  cos  0i  -£4/4  sin  0i  +r5/5  cos  (0i  +  135) 
+£5/5  sin  (0i  +  135)  +^4/4  sin  fa—Xih  cos  0i 
-r5/5  sin  (01+135)  +x5I5  cos  (0i  +  135)].       (264) 

Using  E2  as  the  reference  vector,  76  is  I&  (cos  02  —j  sin  02) 
and  /s  is  1  5  cos  (02+45)—  j  sin  (02+45).  "Substituting 
these  values  in  equation  (263)  and  expanding,  the  value  of 


Vb  =  E2  —7-6/6  cos  02  —XQ!G  sin  02  —  rs/5  cos  (02  +  45) 
—0:5/5  sin  (02  +45)  +j[ro/6  sin  02  —  XQ!Q  cos  02 
+7-5/5  sin  (02  +45)  -zs/5  cos  (02  +45)].  (265) 

In  Fig.  93,  the  vectors  representing  the  drops  are  exagger- 
ated for  the  sake  of  clearness,  and  it  is  evident  that  Va  and 
Vi>  are  not  equal  nor  are  they  90°  apart. 

119.  Three-phase  Connections.  —  In  a  three-phase  ma- 
chine, there  are  three  identical  armature  windings  with 
corresponding  inductors  in  each  winding  spaced  120  elec- 
trical degrees  apart.  The  e.m.f's  in  these  windings  are 
therefore  120°  apart  in  phase  when  the  positive  directions 
are  so  chosen  as  to  be  in  the  same  direction  across  the  arma- 
ture face  in  each  of  three  corresponding  inductors,  one  in 
each  winding,  spaced  120°  apart.  In  practice,  these  wind- 
ings are  connected  according  to  one  of  two  different  schemes. 


POLYPHASE  CURRENTS 


177 


One  is  known  as  the  "  delta  "  connection,  generally  pic- 
tured as  shown  in  Fig.  94,  and  the  windings  are  so  con- 
nected that  the  positive*  directions  in  the  three  windings 
are  in  the  same  direction  around  the  delta.  The  other  is 


known  as  the 


star"  or  "7' 


connection,   generally  pic- 


tured as  shown  in  Fig.  97  and  the  windings  are  so  connected 
that  the  positive  directions  in  the  three  windings  are  in 
the  same  direction  with  respect  to  the  common  junction. 
In  the  delta  connection  the  three-line  wires  are  connected 
to  the  three  points  where  the  windings  join  each  other,  and 
in  the  Y  connection,  the  line  wires  are  connected  to  the 
three  free  ends  of  the  windings. 

Line  a 


Line  b 


Line  C 


FIG.  94. 

120.  Relation  of  Line  Voltages  to  Phase  Voltages  in 
Three-phase  Delta-connected  Systems. — In  Fig.  94,  the 
arrows  EI,  E2, 'and  £3  represent  the  chosen  positive  direc- 
tions of  the  voltages  generated  in  phase  1,  2,  and  3  respec- 
tively. It  is  evident  that  the  line  voltages  are  numer- 
ically the  same  as  the  phase  voltages.  If  the  positive  direc- 
tions for  the  line  voltages,  V\,  ¥2  and  F3  are  chosen  as  shown 
in  Fig.  94  then  the  vector  diagram  of  all  voltages  will  be 
as  shown  in  Fig.  95a.  Note  that  the  same  vector  which 
represents  the  voltage  from  terminal  (2)  to  terminal  (1) 
through  the  winding,  also  represents  the  voltage  from  the 
terminal  (1)  to  terminal  (2)  across  the  line;  if  the  positive 
direction  across  the  line  had  been  chosen  in  the  opposite 
direction  from  that  shown  in  Fig.  94,  then  the  vector  dia- 
gram would  have  been  as  shown  in  Fig.  956. 


178 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


The  vector  sum  of  the  three  voltages,  as  may  be  seen 
from  the  vector  diagram,  is  zero  and  therefore  the  result- 
ant voltage  around  the  delta  is  zero.  That  this  relation 
holds  at  every  instant  may  be  shown  mathematically  as 
follows:  Let  ei,  e2  and  63  be  the  instantaneous  values  of  the 
three  e.m.f  s  then 

ei=E'i  sin  *>?,  (266) 

e2  =  Ef2  sin  (orf  - 120)  (267) 

e3  =  E'3  sin  ( orf  -  240)  (268) 

#3 


£"2  and 


FIG.  95a. 


FIG.  956. 


where  the  prime  (')  indicates  maximum  values.     The  sum 
of  these  is,  assuming  the  maximum  values  to  be  equal, 


ei+e2+ez=E'  [sin  wZ-fsin  (o>« - 120) +sin  (ut-240)].   (269) 
Expanding  the  quantity  in  brackets,  we  get 

sin  co£+sin  uZ  cos  120  —cos  wZ  sin  120  +sin  &t  cos  240  — 


(270) 


1   .  V3  i   . 

cos  w^  sin  240  =  sin  wZ  —  ^  sin  w^ — -^-  cos  wZ  —  ^  sin 


cos  w= 


121.  Relation  of  Line  Currents  to  Phase  Currents  in 
Three-Phase  Systems. — (a)  Delta  Connection. — Assume  that 
the  loads  are  balanced ;  that  is,  that  the  three  currents  in  the 


POLYPHASE  CURRENTS 


179 


phases  are  equal  in  value  and  are  120°  apart  in  phase 
relation.  Choose  positive  directions  as  shown  in  Fig.  94. 
Then  the  vector  diagram  will  be  as  shown  in  Fig.  96.  The 
current  in  line  a  is  the  vector  sum  of  Ii  and  —Is',  in 
line  b  it  is  the  vector  sum  of  1 2  and  — /i;  and  hi  line  c  it 
is  the  vector  sum  of  Is  and  — 12.  The  statements  made  in 
the  preceding  sentence  are  true  whether  the  circuits  are  bal- 
anced or  not.  In  the  case  of  balanced  circuits,  however, 
it  is  evident  from  the  vector  diagram  that  each  line  cur- 
rent is  30°  and  150°  respectively  behind  the  two  currents 


FIG.  96. 

which  enter  that  line.  It  is  also  evident  that  the  value  of 
the  line  currents  is  equal  to  2  cos  30°  times  the  value  of 
the  phase  currents;  that  is,  the  line  currents  areV3  times 
the  phase  currents. 

These  relations  are  shown  mathematically  as  follows: 
Let  ii,  12  and  is  be  the  instantaneous  values  of  the  currents 
and  let  the  maximum  values  be  indicated  by  //,  I2',  and 
73';  then 

t'i  =  /i'sin(o<  (271) 

12=72' sin  ( co* -120),  (272) 

i*3  =  /3'sin  (c^-240).  (273) 

The  instantaneous  value  of  the  current  in  line  a  is  then 

ia  =  i!  -i3  =  TV  sin  ut  - 1*  sin  (&>$  -240).  (274) 


180  ELECTRIC  AND  MAGNETIC  CIRCUITS 

Assuming  the  maximum  values  to  be  equal 

ia  =  /'[(sin  o><  -sin  (ut  -240)],  (275) 

=  /'  (sin  wZ-sin  ut  cos  240-fcos  ut  sin  240)      (275a) 


=        sn  (o         gn  G)~~       cos 


\ 

V>  (2756) 

(3  ^3  \ 

2  sin  u£—  -o-  cos  o)£  ),  (275c) 


sin  wj_    cos  orf,  (275d) 

'  (sin  w£  cos  30  -cos  w£  sin  30),  (275e) 

'sin  (<o*-30).  (276) 


The  value  of  the  line  current  is  therefore  equal  to 
times  the  value  of  the  phase  current,  and  the  line  current  Ia 
lags  30°  behind  the  phase  current  /i.  The  values  of  the 
other  line  currents  can  be  found  similarly  and  the  relations 
will  be  the  same. 

In  the  case  of  unbalanced  currents  the  line  currents 
would  be  calculated  as  follows,  where  0i  is  the  angle  between 
1  1  and  73,  62  is  the  angle  between  72  and  I\  and  63  is  the  angle 
between  h  and  1  2: 


cos  (180-0i)-.//3  sin  (180  -0i),        (277) 
cos  (180-02)  -j7i  sin  (180  -02),       (278) 
L  =  J3  +  /2  cos  (180  -  03)  -j72  sin  (180  -  03).        (279) 

(6)  Y-Connection.  —  Evidently  the  line  current  and  the 
phase  current  are  identical  in  this  method  of  connection. 
See  Fig.  97. 

122.  Relation  of  Line  Voltages  to  Phase  Voltages  in 
Three-phase  Y-connected  Systems.  —  Assume  that  the  volt- 
ages are  balanced;  that  is,  equal  in  value  and  120°  apart  in 
phase.  Choose  the  positive  direction  shown  in  Fig.  97. 


POLYPHASE  CURRENTS 


181 


Then  the  vector  diagram  will  be  as  shown  in  Fig.  98.  The 
line  voltage  V\  is  the  vector  sum  of  E\  and  —  E%',  Vz  is  the 
vector  sum  of  E%  and  —  £3;  and  ¥3  is  the  vector  sum  of 
Ez  and  -Ei.  This  statement  is  true  whether  the  voltages 
are  balanced  or  not;  but  in  the  case  of  balanced  voltages, 
it  is  evident  from  the  vector  diagram  that  the  line  voltages 
are  30°  and  150°  respectively  ahead  of  the  two  voltages 
which  combine  to  make  them.  It  is  also  evident  that  the 
line  voltages  are  equal  to  ^3  times  the  phase  voltages. 
These  relations  may  be  found  mathematically  by  the 


Line  a 


k' 

kn 

Line  b 

I, 

Line  c 

FIG.  97. 


V, 

FIG.  98. 


same  method  as  was  used  for  finding  the  line  currents  in  the 
case  of  the  delta  connection. 

193.  POWCJ  in  Three-phase  Circuits. — The  total  power  in 
any  three-phase  circuit  is 

P=EJi  COS  01+^2/2  COS  02+^3/3  COS  03,         (280) 

where  E\,  E2,  and  E%  are  the  phase  voltages,  /i,  1 2,  and  7s 
are  the.  phase  currents,  and  cos  <£i,  cos  <£2  and  cos  </>3  are  the 
corresponding  power  factors  in  the  three  phases.  If  the 
circuits  are  balanced,  however,  then  Ei=E2  =  E3, 11=I2  =  73 
and  cos  4>i=cos  tf>2=cos  ^3  and 

P=3EIcos<f>y  (281) 

where  E,  I,  and  cos  <j>  are  the  common  values  of  the  phase 
volta£3,  current  and  power  factor.  The  power  may  be 
expressed  in  terms  of  line  voltage,  F,  and  line  current  /', 


182  ELECTRIC  AND  MAGNETIC  CIRCUITS 


by  substituting  F/3  for  E  in  the  case  of  a  F-connection, 
or  I'/^3  for  7  in  the  case  of  a  delta  connection.  The  power 
would  then  be  expressed  as 

P  =  V3~F7'cos  0.  (282) 

It  is  to  be  particularly  noted  that  in  this  expression 
the  angle  4>  is  the  power  factor  angle  of  the  phases  and  is  not 
the  angle  between  line  voltage  and  line  current,  although 
V  and  I'  are  line  voltage  and  line  current  respectively.  In 
the  delta  connection  V  =  E  and  in  the  F-connection  I'  =  I, 
so  that  the  expression  is  correct  for  either  method  of  con- 
nection. 

124.  Power  Measurement  in  Three-phase  Circuits.— 
The  total  power  delivered  to  a  polyphase  system  may  of 
course  be  measured  by  connecting  a  single-phase  wattmeter 
in  each  phase  of  the  system  and  taking  the  sum  of  the  read- 
ings. The  connections  for  this  method  applied  to  a  three- 
phase  system  are  shown  in  Fig.  99.  The  sum  of  the  three 


FIG.  99. 

wattmeter  readings  will  give  the  true  power  regardless  of 
condition  as  to  balance  or  wave  form.  If  the  load  is  known 
to  be  balanced,  one  wattmeter  is  sufficient  and  the  total 
power  is  three  times  this  wattmeter  reading;  hi  general, 
however,  load  is  not  well  enough  balanced  to  permit  this. 
In  practice  the  most  common  method  of  measuring  three- 
phase  power  consists  of  using  two  wattmeters  connected 
as  shown  in  Fig.  100,  and  it  will  now  be  shown  that  the  alge- 
braic sum  of  the  readings  of  these  two  wattmeters  gives  the 


POLYPHASE  CURRENTS 


183 


correct  total  power  regardless  of  balance,  wave  form  or  power 
factor.  A  wattmeter  registers  the  average  value  of  the 
product  of  the  instantaneous  values  of  the  current  through 
its  current  coil  and  the  p.d.  across  its  potential  circuit; 
the  average  value  of  this  product  is  the  average  power  devel- 
oped in  the  circuit  in  which  the  wattmeter  is  connected. 
(See  Article  93.)  If  the  positive  directions  of  current  in 
the  current  coil  and  p.d.  in  the  pressure  circuit  are  both 
chosen  in  the  same  direction  with  reference  to  the  common 
point  of  connection,  (point  p  in  Fig.  100),  the  wattmeter 
will  read  positively  (that  is,  forward  on  its  scale)  when  the 
equivalent  sine  waves  of  current  and  p.d.  are  less  than  90° 
out  of  phase;  it  will  read  negatively  (that  is,  backward) 


Line  a 


Line  C 


Line  b 


W 


FIG.  100. 


when  these  waves  are  more  than  90°  out  of  phase.  In  the 
latter  case,  it  may  be  made  to  read  positively  by  reversing 
the  connections  of  either  coil;  in  practice,  the  potential 
coil  is  reversed  under  such  circumstances.  In  Fig.  100  let 
the  positive  directions  be  taken  as  shown  by  the  arrows 
and  let  the  small  letters  indicate  the  instantaneous  values 
of  current  and  p.d.  Then  the  reading  of  wattmeter  No.  1 
will  be 

Wi  =  average  (eiia),  (283) 

and  the  reading  of  wattmeter  No.  2  will  be 

Wz  =  average  ( —  e^ib) .  (284) 

The  sign  of  e2  is  negative  because  the  positive  direction  of 
62  is  chosen  to  be  from  line  c  to  line  b  but  a  positive  reading 


184  ELECTRIC  AND  MAGNETIC  CIRCUITS 

on  the  wattmeter  requires  that  the  positive  direction 
through  the  pressure  circuit  shall  be  from  line  6  to  line  c. 
The  sum  of  these  two  readings  is 

Wi  +W2  =  average  (eiia)  -haverage  (  -e2ih).       (285) 
But 

ia  =  ii  —  1*3,  and  ib  =  i3  —  i2  ; 
therefore 

Wi  +  W2  =  average  (e\ii)  —average  (621*3) 

—average  (eii'3)+  average  (621*2).  (286) 
But 

61+62=  —63] 
therefore 

Wi+W2  =  average  (e  it  i)  +  average  (621*2) 

+  average  (e3i*3)  ,     (287) 

which  is  the  total  average  power  delivered  to  the  three 
phases. 

If  the  effective  values  of  current  and  p.d.  be  used,  the 
wattmeter  readings  will  be 


,  (288) 

and 

W2=E2Ibcos(3,  (289) 

where  a  is  the  phase  angle  between  E\  and  Ia,  and  ft  is  the 
phase  angle  between  —  E2  and  /&.  These  pjiase  relations 
are  shown  in  Fig.  101,  which  is  the  vector  diagram  for 
Fig.  100.  It  will  readily  be  seen  that  if  'fa  or  <£3  or  both 
are  increased,  a  will  be  increased  and  may  become  equal  to 
or  greater  than  90°.  See  Fig.  102.  If  a  becomes  equal  to 
90°,  the  reading  of  wattmeter  W\  will  become  zero;  if  a 
becomes  greater  than  90°,  then  the  wattmeter  will  read 
backward  or  negatively,  and  its  connections  must  be 
reversed  in  order  to  get  the  value  of  this  negative  reading. 
The  total  power  will  then  be  the  numerical  difference  of  the 
two  wattmeter  readings.  Therefore  when  two  wattmeters 
are  connected  in  a  three-phase  circuit  in  which  the  power 
factors  are  unknown,  or  are  known  to  be  low,  and  the 


POLYPHASE  CURRENTS 


185 


pressure  coils  are  connected  so  that  both  meters  read  up  on 
their  scales,  there  will  be  uncertainty  as  to  whether  the  read- 
ings should  be  added  or  subtracted.  To  determine  which  to 
do,  the  load  may  be  switched  off  and  a  load  which  is  known  to 
be  non-inductive  (incandescent  lamps,  for  example)  put  in  its 
place;  then  if  both  meters  read  up  on  their  scales,  their 
readings  on  the  original  load  are  additive,  but  if  one  watt- 
meter reads  backward,  the  original  readings  must  be  sub- 
tracted. Generally,  unless  the  loads  are  considerably  un- 
balanced, the  wattmeter  giving  the  smaller  reading  is  the 


FIG.  101. 

one  in  doubt  and  the  following  more  simple  method  will 
show  whether  its  reading  is  to  be  added  or  subtracted; 
disconnect  the  potential  terminal  from  the  common  wire 
(line  c  in  Fig.  100)  and  connect  it  to  the  line  in  which  the  other 
meter  is  connected.  If  the  wattmeter  reads  backward  its 
original  reading  is  negative.  For  example,  let  Fig.  102 
be  the  vector  diagram  of  a  certain  load,  connections  being 
as  shown  in  Fig.  100.  Ia  is  more  than  90°  behind  EI  and 
to  get  a  forward  reading  on  W\  its  potential  circuit  must  be 
connected  so  that  the  voltage  on  it  is  —E\]  but  this  fact  is 
not  known  until  the  test  has  been  made.  If  the  terminal 


186 


ELECTRIC  AND  MAGNETIC  CIRCUITS 


q  of  Wi  be  taken  from  line  c  and  connected  to  line  b,  the 
voltage  on  the  potential  circuit  will  be  E3  and  the  angle 


FIG.  102. 


between  E$  and  Ia  is  more  than  90°  so  that  the  reading  will  be 
backward.  If  a  had  been  less  than  90°  the  voltage  on  the 
potential  circuit  of  W\  (for  a  forward  reading)  would  have 


-E, 


-0) 


FIG.  103. 


been  Ei  and  when  q  was  carried  to  line  6,  the  voltage  on  the 
potential  circuit  would  have  been  -E3  and  the  reading 
would  not  have  reversed.  Therefore,  a  reversed  reading 


POLYPHASE  CURRENTS 


187 


when  this  test  is  made  indicates  that  the  difference  of  the 
two  wattmeter  readings  is  the  true  power. 

In  the  special  case  of  balanced  load  and  power  factor,  the 
vector  diagram  of  Fig.  103  applies.  The  reading  of  one 
wattmeter  will  be 

Wi=EJa  cos  (30  +  0),  (290  J 

and  the  other 

W2  =  E2Ib  cos  (30  -  0).  (291) 

When  the  power  factor  becomes  0.5,  <£=60°  and  TFi=0, 
while 


cos  60°  = 


vt 


FIG.  104. 

where  E  and  7  are  the  phase  voltage  and  current,  re- 
spectively. With  power  factors  less  than  0.5  wattmeter  W\ 
will  read  negative. 

In  the  case  of  balanced  circuits  the  power  factor  of  the 
load  may  be  computed  from  the  two  wattmeter  readings  as 
follows  : 

Wi  +W2  =  EJa  [cos  (30  +  0)  +cos  (30  -  <£)] 


,     (292) 

W2-Wi=EJa  [cos  (30-0)  -cos  (30  +  0)] 

=  EJa  sin  0.     (293) 
Therefore 


188  ELECTRIC  AND  MAGNETIC  CIRCUITS 


FIG.  105. 


FIG.  106. 


POLYPHASE  CURRENTS  189 

125.  Line  Drop  in  Three-phase  Circuits. — Fig.  105 
is  the  vector  diagram  showing  the  relations  of  the  line  drops 
to  the  voltages  at  the  two  ends  of  a  transmission  line  as 
represented  in  Fig.  104.  To  the  student  is  left  the  problem 
of  formulating  the  equations  for  computing  the  voltages  at 
the  generating  end  when  the  constants  of  the  line  and  the 
voltages,  currents  and  power  factors  of  the  load  are  given. 
The  principles  are  the  same  as  those  used  in  connection 
with  the  two-phase  problem  at  the  end  of  Article  118. 
In  the  special  case  of  balanced  load  and  power  factor,  the 
generator  voltage  is 


Vi  =  74  cos  0  +  v3r«7«-h;  (F4  sin  4>+^3xaIa).     (295). 

This  formula  may  be  deduced  directly  from  the  vector  dia- 
gram in  Fig.  106. 


INDEX 


Abampere,  definition,  21 
Abhenry,  definition,  88 
Abohm,  definition,  26 
Abvolt,  definition,  28 
Acceleration,  1 
Admittance,  140 

symbolic  expression  for,  144 
Alternating  current,  117 

average  value  of  sine  wave,  123 

effective  value  of  sine  wave,  122 

in  capacity  only,  127 

in  inductance  only,  124 

in  resistance  only,  123 

in  R,  L,  and  C  in  series,  132 
Alternating  current  waves: 

analysis  of  non-harmonic,  157  ff. 

average  value  of  non-harmonic,  165 

effective    value    of  non-harmonic, 
164 

equation  for  sine  wave,  118 

equation  for  non-harmonic  wave, 

154 

Alternating  currents  in  parallel,  150 
Alternating  e.m.f.,  117 

generation  of,  118 
Alternating  e.m.f  s  in  series,  145 
Alternations,  definition,  120 
Ammeters,  55 
Ampere,  a  rate  of  flow,  9 

definition,  21 

international  standard,  22 
Ampere-hour,  23 
Ampere- turn,  67 
Analogy  of  capacity  circuit,  129 
Analogy  of  inductive  circuit,  125 
Angle  of  phase  difference,  121 
Angular  velocity,  118 
Apparent  power,  138 


Average  value: 

of  non-harmonic  wave,  165 
of  sine  wave,    123 

B-H  curves,  71,  73 
Battery  circuits,  31 

Capacity,  electrostatic,  103 

of  parallel  plate  condenser,  104 

of  transmission  line,  105 

short-circuited    with    inductance, 

115 
Cells: 

in  parallel,  34 

in  series,  33 

storage,  30 

voltaic,  29 
Charge,  electric,  99 
Charging  current,  107 
Circuits,  electric,  solution  of: 

mixed  a.c.,  151 

mixed  d.c.,  45 

parallel  a.c.,  149 

parallel  d.c.,  43 

series  a.c.,  144 

series  d.c.,  42 

Circuits,  magnetic,  solution  of,  71,  75 
Circular  mil,  39 
Condensers,  103 

capacity  of  parallel  plate,  104 

charging  and  discharging  through 
resistance,  112 

energy  of,  107 

in  parallel,  105 

in  series,  105 
Conductance,  43 

in  a.c.  circuits,  141 
Corona,  110 


191 


192 


INDEX 


Coulomb,  9,  23 
Crest  factor,  165 
Current 

alternating,     see    alternating    cur- 
rent 

charging,  107 

definition,  10 

decay  in  inductive  circuit,  91 

displacement,  107 

growth  in  inductive  circuit,  90 

positive  direction  of,  18 

unit  of,  21 
Cycle,  definition,  119 

Delta-connection,  177  ff. 
Density,  magnetic  flux,  68 

electrostatic  flux,  102 
Dielectric,  definition,  100 

constant,  102,  111 

hysteresis,  110 

losses,  110 

strength,  110 

Displacement  current,  107 
Drop,  potential,  28 

internal  resistance,  32 
Dyne,  definition,  4 

Eddy  currents,  83 
Effective  resistance,  134 
Effective  value: 

of  non-harmonic  wave,  164 

of  sine  wave,  122 
Electric  charge,  99 
Electrical  degrees,  120 
Electricity,  8 
Electromagnet,  pull  of,  85 
Electromotive  force,  9 

alternating,  117 

as  work  done,  38 

generation  of,  58-62 

self-induced,  87 

unit  of,  27 
Electromotive  forces: 

in  parallel,  35 

in  series,  33,  145 
Electrostatic  field  intensity,  100 

distribution  of,  108 

units,  102,  103 


Electrostatic  potential,  102 

unit,  102 
Energy,  definition,  5 

of  a  condenser,  107 

of  an  electric  current,  38 

of  a  magnetic  field,  93 
Equivalent  phase  difference,  167 
Equivalent  sine  wave,  167 
Erg,  definition,  5 

Farad,  definition,  103 
Field,  electrostatic,  100 

energy  of,  107 

unit,  101 
Field,  magnetic,  10,  12 

action  on  a  wire,  20,  22 

around  a  wire,  17 

energy  of,  93 

of  a  solenoid,  19 

positive  direction  of,  18 
Field  intensity,  electrostatic,  100 
Field  intensity,  magnetic,  12,  67 

around  a  straight  wire,  64 

at  center  of  large  coil,  65 

in  a  solenoid,  76 
Flux,  magnetic,  13 
Flux  density,  magnetic,  68 

electrostatic,  102 
Flux-linkage,  58 
Force,  definition,  3 

between  magnet  poles,  11 

between  two  parallel  wires,  64 

on  a  wire  in  a  magnetic  field,  22 

units  of,  4 
Form  factor,  165 
Frequency,  definition,  119 

of  resonance,  149 

Galvanometer,  23 
Gauss,  68 
Gilbert,  66 
Gradient,  potential,  108 

Henry,  definition,  88 
Hysteresis,  78-82 

Impedance,  134 
symbolic  expression  for,  143 


INDEX 


193 


Inpedances  in  parallel,  149 

in  series,  144 
Inductance,  87,  88 

formula  for  air-cored  coils,  89 

mutual,  97 

of  two  long  parallel  wires,  94 
Intensity  of  electrostatic  field,  100 

distribution,  108 

units,  102,  103 
Intensity  of  magnetic  field,  12,  67 

around  a  straight  wire,  64 

at  center  of  large  coil,  65 

in  a  solenoid,  76 

unit  of,  12 

Joule,  definition,  5 
Joule's  Law,  26 

Kilovolt-ampere,  139 

Kirchhoff's  Laws,  42 

applied  to  d.c.  circuits,  47-50 
applied  to  a.c.  circuits,  171 

Lag,  angle  of,  121 
Lead,  angle  of,  121 
Leakage,  magnetic,  78 

coefficient,  78 
Left-hand  rule,  21 
Lenz's  law,  60 
Line  drop: 

effect  in  two-phase  circuits,  175 

effect  in  three-phase  circuits,  189 
Lines  of  force,  meaning  of,  12 

properties  of,  13,  15 

from  unit  pole,  63 
Linkages,  flux,  58 

Magnetic  circuits,  solution  of,  71,  75 
Magnetic  field,  10,  12 

action  on  a  wire,  20,  22 

around  a  wire,  17 

energy  of,  93 

of  a  solenoid,  19 

positive  direction  of,  18 
Magnetic  flux,  13 
Magnetic  leakage,  78 
Magnetic  lines,  properties  of,  13,  15 


Magnetism,  10 

modern  theory,  14 
Magnetization  curves,  71,  73 
Magnetizing  force,  67 

units  of,  67 
Magnetomotive  force,  66    . 

units  of,  66,  67 
Magnets,  10,  11 
Mass,  definition,  2 

standard  of,  2 
Maxwell,  1-3 
Mho,  44 
Microfarad,  104 
Mil,  39 
Mil-foot,  40 
Milli-henry,  88 
Mutual  induction,  97 

coefficient  of,  98 

Non-harmonic  waves,  154 
Non-inductive  circuits,  92 

Ohm,  definition,  27 

international  standard,  27 
Ohm's  law,  27 
Oscillograph,  156 

Peak  factor,  165 
Period,  119 
Permeability,  68 

curves,  70 
Phase,  121 

Phase  difference,  angle  of,  121 
Pole,  unit,  11 
Potential,  9 

fall  of,  28,  32 

rise  of,  28 

Potential  difference,  9,  28 
Potential  gradient,  108 
Potentiometer,  53 
Power,  definition,  7 

apparent,  138 

component  of  current,  140 

component  of  e.m.f.,  139 

in  electric  circuits,  36 

in  non-harmonic  a.  c.  circuits,  165 

in  sine  wave  a.c.  circuits,  135 

in  three-phase  circuits,  181  ff. 
Power  factor,  138 


194 


INDEX 


Quantity,  electric,  9,  23 

Reactance 

capacity,  128 

inductive,  127 

positive  and  negative,  134 

with  non-harmonic  waves,  167,  169 
Reactive  factor,  139 
Reactive  power,  139 
Reluctance,  69 
Resistance,  nature  of,  26 

effective  hi  a.c.  circuits,  134 

specific,  40 

standard  of,  27 

temperature  coefficient  of,  41 

unit  of,  26 
Resistances: 

in  mixed  circuits,  45 

in  parallel,  43 

in  series,  42 
Resistivity,  40 
Resonance,  147 
Right-hand  rule,  63 

Self-induction,  87 
Sine  wave,  119 

equivalent,  167 
Skin  effect,  96 
Slide-wire  bridge,  52 
Solenoid,  19 

field  intensity  in,  76 

inductance  of,  88 
Specific  inductive  capacity,  102 
Specific  resistance,  39 
Star  connection,  177,  ff. 
Storage  cells,  30 
Susceptance,  141 
Symbolic  method,  141  ff. 

Temperature: 

coefficient,  41 

effect  on  resistance,  40 
Three-phase  connections,  176 

vector  diagrams,  177,  ff. 


Three-wire  circuit,  50 
Transmission  line: 

capacity  of,  105 

inductance  of,  94 
Two-phase  connections,  172 

vector  diagrams,  175 
Two-wattmeter  power  measurement, 
183 

Unit  pole,  11 
Units: 

of  current,  21 

of  e.m.f.,  27 

of  electrostatic  capacity,  103 

of  electrostatic  field  intensity,  103 

of  force,  4 

of  inductance,  88 

of  magnetic  field  intensity,  12,  68 

of  magnetic  flux,  13 

of  magnetic  flux  density,  68 

of  magnetomotive  force,  66 

of  power,  7,  36,  139 

of  quantity,  23 

of  resistance,  26 

of  work  and  energy,  7,  37,  38 

Vector  representation  of  alternating 

quantities,  130 
Voltage,  32 
Voltaic  cell,  29 
Voltmeters,  55 

Watt,  definition,  7 
Watt-hour,  definition,  7 
Waves,  sine,  119 

equivalent  sine,  167 

non-harmonic,  154 
Wheatstone  bridge,  51 
Wire  size  to  produce  given  m.m.f.,  76 
Work,  definition,  5 

done  in  cutting  magnetic  field,  63 

Y-connection,  177  ff. 


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